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J45.0g Chronische Krankheiten der unteren Atemwege (J40-J47)

regernachweis ergibt Hyemophilus influencae. J G Vorwiegend allergisches Asthma bronchiale. J14 G Pneumonie durch Haemophilus influenzae. B! ICD J Vorwiegend allergisches Asthma bronchiale Allergische: Bronchitis o.n.A. Allergische: Rhinopathie mit Asthma bronchiale Atopisches Asthma. Asthma bronchiale, allergisches; extrinsisches Asthma, atopisches Asthma, exogenes allergisches Asthma bronchiale. ICD Code J Definition Asthma. F54, J, Psychogenes Asthma bronchiale. J, Vorwiegend allergisches Asthma bronchiale. Allergische Bronchitis. Allergische Rhinopathie mit Asthma. Diagnosen: JG Allergisches Bronchialasthma J A Ausgeschlossen: Lungenemphysem JV Verdacht auf Allergische Rhinopathie durch Pollen.

ICD J - Asthma bronchiale, nicht näher bezeichnet. Medikamente und Infos zu ICD Code J J45, Asthma bronchiale. J, Vorwiegend allergisches Asthma bronchiale. J​1, Nichtallergisches Asthma bronchiale. J, Mischformen des Asthma. F54, J, Psychogenes Asthma bronchiale. J, Vorwiegend allergisches Asthma bronchiale. Allergische Bronchitis. Allergische Rhinopathie mit Asthma.

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Acoustic Paradiso - Gibson J45 Standard & J45 Custom Comparison Das schwierige Asthma ist gekennzeichnet durch den Krankheitsverlauf erschwerende Faktoren wie schlechte Compliance, psychosoziale Probleme, wiederholte Exposition gegen Schadstoffe und Allergene oder auch unbehandelte andere Erkrankungen. Stickstoffmonoxid NO ist ein wichtiger Botenstoff im Check this out, der natürlicherweise an unterschiedlichen Stellen J45.0g Körper entsteht. Band 22,S. Jahrgang, Es wird folgendes Vorgehen empfohlen:. Eine Vielzahl von Reizen verursacht die Zunahme der Empfindlichkeit der Atemwege bronchiale Hyperreaktivität oder auch bronchiale Hyperreagibilität und die damit verbundene Entzündung. Diese hat zentrale Bedeutung beim Asthma. Jahrgang, Oft treten beide Reaktionen auf link dual reaction. Juniabgerufen am Die Obstruktion ist die Verengung des Lumens der Atemwege Verkleinerung des verfügbaren Querschnitts infolge von Schleimhaut ödemen Flüssigkeitseinlagerung in die Schleimhautvon vermehrter bzw. Eine Vielzahl von Reizen verursacht die Zunahme der Empfindlichkeit der Atemwege bronchiale Hyperreaktivität oder auch bronchiale Hyperreagibilität und die damit verbundene Entzündung. Augustine Tee, K. Papiris, A. Wirkstoffe von A bis Z Über Beim allergischen J45.0g wird bei Kindern zum Teil manchmal noch auf Cromoglicinsäure, Nedocromil oder Montelukast zurückgegriffen. J45, Asthma bronchiale. J, Vorwiegend allergisches Asthma bronchiale. J​1, Nichtallergisches Asthma bronchiale. J, Mischformen des Asthma. Der ICD Code J45 beschreibt Chronische Krankheiten der unteren Atemwege (​JJ47), konkret Asthma bronchiale. Mischformen des Asthma bronchiale (Kombination aus J und J) Asthma J G Pneumonie durch Klebsiella pneumoniae. EIGENE NOTIZEN. ICD J - Asthma bronchiale, nicht näher bezeichnet. Medikamente und Infos zu ICD Code J ICD J - Vorwiegend allergisches Asthma bronchiale. Medikamente und Infos zu ICD Code J

Solution by an approximate method; see In the usual case, a will be preferred. The remaining parts Of tiiis section summarize some general properties of 1.

They are inserted here for convenience of reference and can be consulted as needed. Analytic Functions. It is the purpose of this section to consider some properties of analytic functions as used in the study of differential equa- tions.

For further details and proofs of the statements which follow, see appropriate references in the Bibliography, for example, Ince-1, Codding- ton and Levinson.

The test for analyticity can be be made simply, for it is only necessary, according to 3 , that the function and its derivatives exist at the point in question.

There are several kinds of singular points. Such a singularity is called a poh of order n or a nonessential singular point. The misbehavior of the function is effectively avoided by use of the multi- plicative term.

If a circle can be drawn with center at the singular point so that no other singular point is enclosed, the singular point is an isolated one.

Within a circle of radius less than 2ln, there are an infinite number of poles. Essential singular points.

It is said to be an essential singular point. Branch points. Whatever the order, some derivatives of finite order and all higher ones will be infinite at a branch point.

Fixed and movable singular points. Examination of the coefficients in a differential equation will reveal the nature of its singular points, which can be of the kinds described in a, b, c.

These are the fixed or intrinsic singular points of the differential equation. It does not follow that solu- tions of the equation will also have singular points of the same kind.

Only under special conditions is the origin a singular point of the solution. Such a singular point, which moves about as the initial values are varied, is a movcMe or parametric singular point.

Singular points for linear equations are always fixed. A nonlinear equation of first order and of first degree can have movable poles and movable branch points but no movable essential singularities.

Nonlinear equations of second or higher order can have movable singular points of all kinds. Movable branch points and essential singularities, excluding poles of finite order, are often called critical points of the differential equation.

There are several cases, depending on the nature of fix, y. Proceed as foUows. Since the latter constant is arbitrary it will be the constant of integration.

For some comments on convergence, see d. The method of undetermined coefficients. Replace the left-hand side of the differential equation by the derivative of 2.

The result, which no longer contains y, is an identity in x. Equate coefficients of equal powers of x in this equation and obtain relations with which Ai, Az,.

There are three possibilities. All coefficients after At are zero so Uiat 2 is a polynomial of degree k. The present procedure can, of course, be used in such cases, if desired.

A general law results for the coefficients in terms of Ao. Such a relation is called a two-term recursion formula or a, first-order difference equation; see also iii.

There is no general law for the coefficients so that three-term for- mulas or even more complicated ones occur. This means that At will usually depend on At-z, etc.

In fact, it may depend in some rather involved way on all of the coefficients which precede it and no explicit solution can be found for A as a function of Aq.

It will become more and more laborious to calculate successive coefficients but, nevertheless, such calculations may be continued as long as desired.

Many-term recursion formulas are linear finite-difference equations. It is often convenient to study them by such methods; see, for example, Jordan, Milne- 1, MOne- Thompson.

Frequently, one wishes a series solution so that y can be determined within some specified limits of error. This will fix the number of coeffi- cients which must be calculated; see also d.

It should be noted that the existence theorems guarantee that the solution is valid but it is not possible to make tests for convergence, as in ii, since the general term is unknown.

Expansion in a Taylor series. The method is equivalent to that of a, which in the usual case will be easier to apply. Use this result, together with and fy, to calculate y" in 3.

Ciontinue in this way to find the third derivative and as many more terms as may be wanted. Evaluation of the successive derivatives can become quite complicated, if it is necessary to use more than a few terms in the series ; see also d.

The coefficients by integration. For an alternative way of deter- mining the coefficients in the series 2 , repeated integrations may be used rather than differentiations.

For the details, see Convergence of the series solution. If the conditions required by the existence theorem hold, it is certain that 2 is the general solution of 1.

A more exact statement of these conditions may be useful. If further details and proofs are wanted, other sources must be consulted.

Some suitable references are Coddington and Levinson, Goursat, Ince Take Jf equal to or greater than this sum, so that! If t is fixed, the right-hand side of 4 will decrease as x decreases.

It is still possible, in this case, to solve the differential equation. All coefficients vanish at X 0, which means that each term in the right-hand member of the differential equation contains x as a factor.

The Equation of Briot and Bouquet. This is case a of For a generalization of it, see There are three possible situations. Successive values of the At are obtained from 7.

Nonanalytic solutions may also exist ; see c. There are relations between k and the coefficients atj so that 7 can be used to calculate the coefficients At.

The preceding result can be obtained in another way. The last differential equatipa is similar to 5 but the coefficient of u has been reduced by unity.

There are no solutions analytic at the origin but there is a general solution con- taining an arbitrary constant. It is a series in x and x In x; it approaches zero as x approaches zero along a properly chosen path.

For references and further details, see Ince Nonanalytic solutions. The first coefficient cqo is arbitrary.

The series approaches zero as x approaches zero along a properly chosen path. More complete details are given by Goursat, Ince-1, Valiron.

V The Generalized Equation of Briot and Bouquet. It is mMiiiTiftd that both P and Q are convergent double series in x and y, aimiUr to the right-hand member of 5.

However, P and Q are not divisible by any power of x or y. They are described, for example, by Ince 1. Uo is a simple root.

The equation is now of type and can be handled further as described there, b. Uo is a double root. There are two conclusions, both for Jb 0.

The origin is an essential singular point; see lO-I. There are two special cases of interest, which in terms of the original variables, are as follows.

The second equation is of Ricoati type; see Thus 2 is an integral equation which should be solved. Continue in this way and calculate The work is stopped with a solution which is a sufficiently close approxi- mation to the exact solution.

The procedure may become quite tedious, since the integrations are usually more and more difficult to perform. With appropriate restrictions, existence theorems show that the successive integrals converge and that the solution is unique.

It may then be possible to find an approximate solution by one of the following means: a. For these reasons, we limit the descrip- tion of them here.

Graphical methods. Considered geometrically, the differential equation assigns a slope to every point x, y in the x, y-plane. Each will determine a small portion of an integral curve.

The general shape of that curve will become apparent as the plotting is continued. A smooth curve with the line segments as tangents will be the integral curve sought.

This situation is equivalent to the general solution if xq is a constant and yo is a variable parameter or the constant of integration.

Following these general principles, many ingenious variations have been devised. As one possibility, isoclines are first drawn.

These are the loci along which the slope required by the differential equation has a constant value. When a number of isoclines have been drawn, the integral curves can then be sketched on the same plot.

As another possibility, a first approximation to an integral curve could be drawn and this could be improved by redrawing, until two successive curves coincide.

To compute correction. In other cases, a nomograph might be constructed for calculating the corrections. No high order of precision can be expected by gra]thical means but the results may be acceptable for special problems, especially if the graphs are carefully constructed and of sufficiently large size.

Lacking some par- ticular reason for using this method, one of those in c would usually be preferred. For more details about the graphical methods, the book of Levy and Baggott is recommended.

Mechanical methods. Devices for mechanical integration have a long history. Two such instruments are the integraph and the polar planimeter.

The former, described by Abdank-Abakanowicz in , contains a tracing point, which movps along a graph of the integrand.

An attached pen draws the integral curve. The polar planimeter, one model of which was invented by Amsler in , also has a point, which moves along the integrand curve.

A coimected scale and vernier indicates the number of complete or fractional revolutions made in tracing out the perimeter of the integrand curve.

A simple calibration and calculation give the area under 'this curve, which, of course, equals the value of the definite integral.

Either of these devices could be used to solve a simple differential equation in several ways.

For example, the integral equation method, see 12, would give successive approximations to the solution, when the various integrals had been evaluated mechanically.

A wide variety of more sophisticated instruments has been described in the literature and many of them actually constructed.

In some oases, the machine has been invented to solve a particular type of equation, such as that of Bicoati, see 3, or of Abel, see 4.

In other cases, the machine may be more versatile but none of them are simple and all are expensive to build. For many references and further description of such instru- ments, see Kamke.

It was based on addition and integra- tion. The former was achieved by gear boxes and the latter, by a wheel and disk mechanism, similar to that of the polar planimeter.

Originally, a curve was followed manually by the user of the machine. Later, photo- electric curve tracers were added and, eventually, the moving parts were replaced by electronic components.

These developments were a logical outcome of the Mallock electronic machine, which had been invented for solving simultaneous equations.

The modem versions of these machines are called analogue computers. For more details, see Johnson, Soroka. Numerical methods.

The methods of a and b are based on measure- ment and on the properties of smooth curves. The somewhat inexact title of this section suggests that number is now of major importance.

Thus, all operations in these methods are essentially addition or subtrac- tion. They can be carried out by any of the commercially available desk calculating machines or by those high-speed devices known as digital computers.

In general, numerical methods are based on step-by-step integration. The broken-line curve resulting would be a rough approximation to the integral curve desired.

Obviously, the same procedure could be followed with numbers as it is certainly not necessary to draw the curves.

The solution of the differential equation would follow from the Taylor series method of lb. When the derivatives are not easy to obtain, the method could be modified by using interpolation formulas for them.

Alternatively, the integral equations of 12 could be used. If the integration is difficult, use numeri- cal integration.

A slightly different method is that of Bunge and Kutta. In the usual case, it may be the most suitable procedure if a desk machine must be used.

One of these numerical methods may be prefinrred, even though a com- plete solution of the differential equation can be found by anotoer method.

It may then be easier to solve the differential equation numerically than to solve the transcendental equation. Numerical methods are not limited to equations of first order but may be extended to equations of order two or more and to systems of simul- taneous equations.

For some references where more details can be found, see Milne-2, Scarborough. If a digital computer is available for solving the equation, see Wilkes, Wlieeler, and Gill.

If m is a fraction, it will be necessary to rationalize the differential equation and clear it of fractions in order to determine the degree.

However, some of the following methods will apply, even if m is fractional. When transcendental functions of p occur, like Inp or cosp; see Thus, alternative forms of the general solution may be required.

They are described in a, b, c and will be called solutions of types I, II, III, respectively, in the following sections. In addition to the general solution, a singular solution may also exist.

It will satisfy the differential equation but it will not be a special case of the general solution.

When it alone is of interest, go directly to 10, for it may often be obtained without solving the differential equation. Alternatively, a singular solution may also appear in the methods which follow, when the general solution of the equation is sought.

However, the singular solution will usually be lost if common factors are eliminated from both sides of an equation or if they are canceled out in a numerator and a denominator.

When a series solution of the differential equation is desired, or whra the behavior of such a solution is to be investigated in the neighborhood of a singular point, see Al, proceed at once to 7.

Solutions of Type I. It does not need to occur algebraically, however, and in that case, the only re- quirement is that C be arbitrary.

Solutions of Type II. The general solution in the form 1 might be a complicated algebraic or transcendental function. Solutions of Type III.

If the algebra is not too complicated, p could be eliminated between the two simultaneous equations in Ft and and the general solution presented as in a or b.

However, in many cases, such algebraic elimination can be formidable. It is then much simpler, and just as satisfactory, to regard p as a parameter.

To emphasize this meaning for p, it will be replaced by the symbol t. Missing Variables If X, y, or both Ere missing, there are several procedures.

Sometimes, one is easier than the others. For an equation of this type, look at all of the subcases before making a choice of the method to be used.

If a new variable is suggested, see Conversion of the equation into type is sometimes helpful; see Solve for p.

Solve for x. There are two possibili- ties. Use a New Variable. Exchange Variables. The new dependent variable will be X and Ijp s dxidy. It can also be givrni as y x, y, 0 0, which is type I.

Solve for y. Replace pbyt; use 6 and 6 as a parametric solution of type III. Both Variables Missing. Factor it, if possible, to get P -ri p -ra The individual factors can be integrated since the variables are separated in each.

Occasionally, the method of might be useful. Consult 3, 4, 5. Sometunes, the Legoidre transformation, see A1 , will be helpful. The result is p -Fi p -Fa Since each factor is of first degree, the methods of A1 can be used.

Some further information on equations of this case will be found in 8. Differentiate with respect to y.

A first-order equation in y and p is obtained but x is missing. Retain both 1 and 2 as a parametric solution in terms of the parameter t, which has replaced p.

Thus, the solution is type III. Regard y as a function of p. Solve the first-order equation by a method of Al.

Eliminate p between 1 and 3 to get y - -l- C. Differentiate with respect to x. Gall p a variable parameter, rename it t, and give the solution as type III, with both 4 and 6.

Calculate dx Fp dp p -Fg where the subscripts mean partial derivatives; see b. Special cases. If the given differential equation is linear in y, with a constant for its coefficient, the differentiation in a or b can be carried out at once; it is not necessary to solve for y.

Refer to 4. Equations of the special type described there can be solved very easily. Homogeneous Equations and Related Types For the meaning of the word homogeneous, as used here, see Al There are several different kinds of equations.

Proceed in one of the following ways, whichever seems the easiest. Solve fory. Use As another possibility, replace p by t and give a parametric solution of type III.

Introduce a New Variable. The form of the equation may suggest the appropriate transformation. Then, whichever seems easier: i.

Solve for x and proceed according to b. If algebraic difficulties are severe, replace v by the parameter t and give a solution of type III.

The method of is applicable so that. J Restore the original variables or give a parametric solution as in a.

It is type and its solution can be found by a method of that section. Glairauf 8 Equation and Related Types There are three dififerent equations, each quite similar in form.

Glalraut's Equation. It contains no arbit- rary constant; it is not a special case of the general solution 1 ; it satisfies the differential equation.

Alternatively, 1 and 4 may be used as a para- metric solution of type III. In either case, the result is a singular solution ; see 10 for further details.

If y p R 0, the equation is homogeneous; see Make x the dependent variable and p the indepmident variable.

Alternatively, retain 5 and 7 as a solution of type III, replacing p by a parameter t. A singular solution may also exist; see The Legendre transformation.

Alternatively, differentiate 1 with respect to x, solve the resulting second-order equation, see B, to get 2 and use 3 to eliminate one constant.

Change of Variable. Try to convert the differential equation into one of the preceding types by a new dependent variable, a new independent variable, or two new variables.

Specific directions are not readily given but a number of suggestions may result from Al Modifications needed for equations of second or higher degree are usually obvious.

Since an equation of Clairaut t 3 pe, see , is quite simple to solve, seek a transformed equation of that kind, as one possibility. A few special cases of variable transformation are listed in the next section.

Many further examples will be found in Part II. In that case, it may be possible to use: a. An infinite series; see Al A definite integral; see Al- An approximate method; see Al The refierences apply to an equation of the first degree but they may be modified for equations of higher degree.

Since problems of this sort are not common, no further details will be given here. As stated in Al, mathematicians have been interested in finding certain special classes of differential equations which define new tnuosoen- dental functions.

The necessary conditions for equations of degree tvro or greater are stated in 8 and its subsections. With still further restrictions, the special cases of 9 result.

Note that the subscript in 3 does not mean a partial derivative, as it does in 2. There are four cases. Ap vanishes independently of y. Xo vanishes independently of y.

There are singular points of Xi for general values of y. There are singular points for a root of 3. Exclude each of them from subsequent consideration, for we are interested here only in the movable singular points.

Let zo, ya be some point other than those which have been excluded. It may, or may not, be a movable singular point. Series solutions of 1 are wanted in the neighborhood of zq, yo.

There are four special cases. A study of them will reveal the conditions fox a solution with no movable singular points. If that case along is of intexest, go to 9.

Each is an analytic function there. Treat each of the other roots in the same way. The general solution of 1 will be, see Use to find m-1 solutions for the finite roots.

For general values of y there will be ih different roots of p; see Suppose that the roots equal to 9 are pi, p 2 ,. Then, one of the equal roots, pi for example, will either return to its initial value or become equal to one of its partners ps, P8,.

There are several special oases. The differential equation is analytic in x and Y. Its solution is an analytic function; see Al Three possibilities arise in this case.

It is a singular solution; see This solution has a movable branch point. This is the same situation as that in the first case of ii.

The conclusions are similar to those in , except that the multiple root is at infinity. X, y pm-i Other special cases of 8 are presented in 9- Iff.

For further details about the equation, see the following parts of this section. Movable branch points. To avoid them, require that JCq be a function of X alone.

A singular solution. A restriction on a. Further restrictions. Suppose also that P is a many valued function of u.

It is required that tt be a factor of both fm and fm-u t. A branch of order a. It is necessary, according to c, that a - 1 C k.

No movable essential. This is true for 1 , independ- dent of the presence or absence of movable branch points. There can be no fixed singular points, except possibly one at infinity, since the coefficients do not depend on x.

If all conditions of 9 are met, there can be no movable branch points. Furthermore, there are no essential singularities for finite X.

The point at infinity can be an essential singular point, but not a branch point. This is a special case of the general equation in 9.

It is assumed to be irreducible, see 8, and to satisfy all further re- quirements of 9. If uninterested in the details, go to , where these permitted oases are listed.

If the arguments lor 3 are wanted, see the following. If the degree is less than 2m, there are two possibilities, but both can be reduced to the case where the degree is 2m.

Por convenience, the degree of Xlx, y will be taken as exactly 2m. The degree of Xix, y is less than 2m, but it does not contain y as a factor.

It has become of degree 2m in the variables u and x. The degree of X x, y is less than 2m, but it contains y as a factor.

There are equal roots. This means that Ap a;, y s 0; see and When the differential equation of is restricted so that there are no movable branch points, the equation of this section results.

The results are listed in a. There is only one possible differential equation; see b. The Binomial Equations.

If all conditions in are met, the resulting equations are called binomial. For a method of solving each differential equation, see Part II.

The oases of a. There are six possible types ; see Table 1. It becomes type II, if is a constant, not equal to 02 or Degenerate cases.

Each will give one or more degenerate cases of degree lower than 2m. The possible cases are listed in Table 2. TABLE 2.

It satisfies the differential equation. It contains no arbitrary constant. It cannot be obtained by assigning a particular value to the integra- tion constant in the general solution of 1.

Such a function is a aingvJar aolvJtion. Its properties are frequently of interest and, in some problems, it is wanted rather than the general solution of the differential equation.

The equation is reducible. There will be no singular solution. A check on this situation can be made, for the p-discriminant, see , will vanish identically.

The equation is irreducible. A singular solution may exist. In such a case either or could be used. The former may be simpler, since the general solution of the differential equation need not be known.

However, it would be preferable to complete the work of both and , in that order. Then, refer to for a more complete treatment of the problem.

The p -discriminant. Note that the subscript in this relation does not mean a partial derivative. Clear the p-discriminant relation of fractions and radicals, discard any constant factors, but do not reject any functions of the variables.

The equation will then usually be a product of two or more functions of x and y. Test each to see if it satisfies the differential equation.

If it does, it is either a singular solution or a special case of the general solution. Any factor which does not satisfy the differential equation describes a curve related to the general solution of 1.

There are two modifications of the general procedure which may be helpful. The C-dlscrimlnant. The procedure of this section can be used as an alternative or a supplement to that of In either case, the general solution of 1 is first required.

Note, that C here does not mean a partial derivative. The C-discriminant relation will usually be a product of two or more factors.

If this section has been preceded by the work of , the singular solution found there will again appear in Ac. The present section thus furnishes a check on the earlier calculations.

If a factor in Ac fails to satisfy the differential equation, it may or may not duplicate a similar function found by For further information about such functions, see It is often convenient to calculate Ac as follows.

Singular Solutions and Associated Curves. There is no generally accepted definition of a singular solution and at least three have been used by different writers: i.

A function which' satisfies the differential equation but which cannot be obtained by assigning a special value to the arbitrary constant in the general solution.

An envelope to the family of curves given by the general solution of the differential equation. A solution of the equation which occurs in Ay.

In this book, definition 1 is used. If conclusions followed are compared with those in other books, discrepancies may occur unless the same definition has been used in each case.

Each will generaUy be a product of two or more functions. Its factors may describe ouires of three different kinds: a singular solution, a particular solution, a function which does not satisfy the differential equation.

These three oases are conveniently discussed in geometric terms. If uninterested in the details, skip the rest of this section and go to Let the given differential equation be 1 and let its general solution be 4.

If 1 is of first order and first degree, its solution describes a fomily of pUme curves. A single curve of this family is completely identified when a definite value is assigned to the constant C.

Singular points, where the slope becomes indeterminate, require special treatment; see Al Otherwise, only one curve of the family will pass through any chosen point and there wiU be a unique slope at that point.

In algebraic terms, both 1 and 4 are pol 3 momials of degree m, the first in the variable p and the second in C. Each of these polynomials must have m roots but there need not be m different roots in either case, so there may be less than m curves or slopes at a selected point.

An indeterminate slope at a particular point is tem- porarily excluded; see d and e for sqch oases. Equality of roots is recog- nized in algebra by the vanishing of the discriminant.

An alternative method, based on calculus, has also been given in lb. The existence of singular solutions and the other functions that might have arisen by the methods of and depends on the unusual behavior when thel:e are two or more equal roots for p and C.

This is the reason for calculating both discriminants. The possible consequences are described in the following sections.

The envelope. If a curve is tangent to some member of the family 4 at every point, it is the enveJope of the family.

The slope of the envelope is the same as the slope of the integral curves at the points of common inter- section.

A partioalar solution. Occasionally, a particular solution of the differential equation will appear in both disoiiminantB.

At the same time, a special curve may intersect all other members of the family at the same point, so that an infinite number of curves meet there- A smaller number of curves than usual will pass through any other point on the particular curve.

The equation for this special curve will thus appear in both and Ac and, in fact, three times in the formeir bitt only once in the latter.

Such properties often make it possible to identify the situation from the two discriminants ; see also 1. Confirmation of the conclusion, of course, comes from the fact that the particular solution is fixed by some special value of C in the general solution.

As explained in a, a particular solution which is also an envelope is not regarded here as a singular solu- tion. Tac locus.

Suppose that two curves of a family are tangent to each other at some point. This means that there will be two equal values of p, a fact which will be revealed if is examined.

On the other hand, Ac will show nothing about this behavior since the proper number of curves pass through the point of intersection.

Such a point is called a tac point ; the locus of these points is a tac locus. It should be noted that there may be three types of tac points: the two touching curves can have the same curvature ; they can have opposite curvatures ; a point of inflection can occur for one curve at the tac point.

As in other cases, the tac locus can play more than one role. Thus, it could also be eui envelope but, in that case, we would not consider it to be a singular solution.

If the same integral curve is determined by two differ- ent values of C, the equation of this curve will also be a tac locus. More- over, a nodal locus, see d, of the slopes will generally be a tac locus of the integral curves.

Nodal locus. In some cases, a singular point may occur on each of the integral curves. If there are k tangents to the curve at some point, it is a multiple point of order k.

In such a case, two consecutive curves of the integral family will intersect at three differ- ent points.

In the limit when consecutive curves approach coincidence, two points of intersection approach the crossing point of the curves and a nodal locus results!

The nodal locus will appear twice in Af, corresponding to the two branches of the curve. The tangents of each are different, so the nodal locus will not occur in A,,.

In unusual cases, it is also possible for the nodal locus to be an envelope, thus also a singular solution.

As explained in c, a tac locus commonly exists when the integral curves contain nodes. Cuspidal locus. Suppose that the three points of intersection in d coincide.

The position of the singular point is determined as in d, but with tf x, y - 0. There are two real, equal values of the slope and the cmve recedes from the jjoint of tangency in one direction.

The two branches of it, however, are on opposite side of the common tangent. The cuspidal locus will appear three times in Ac, since three loci coincide there.

It will normally ai pear only once in Ap, because there are two equal values of the slope. Sometimes, but not in the usual case, the cuspi- dal locus may also be an envelope, thus a singular solution.

Summary on Singular Solutions. It is assumed that the calculations of both 1 and have been performed. Two discriminant relations will thus be available.

Each will usually be a product of two or more functions. Compare Ap with Ac and it will often be true that both contain one or mord factors.

If a function belongs to more than one category, it will be repeated the proper number of times. The following comments may be helpful.

Do not cancel out any functions of the variables in calculating the discriminants. Constant factors, however, may be discarded. A singular solution is an unusual case, rather than the general case.

In order for fi x to be a solution of the differential equation, hence a singu- lar solution, it is necessary that fzix — dfijdx. Such a relation cannot be expected to hold in general.

No cuspidal or nodal loci can occur for a differential equation which has families of straight lines or conic sections for a general solution.

The general solution of a differential equation can often be given in several different equivalent forms. It sometimes happens that the C-discriminant will give the singular solution with one form, but not with another.

This difficulty would be avoided if both discriminants were studied. The two equations of this section are said to give correct results always if the degree of the differential equation is two or three.

A number of examples, where they appear to give incorrect conclusions, are discussed by Piaggio. Some of his examples have degree greater than three; some have degree of three or less.

Note, however, that he regards any envelope as a singular solution, which is equivalent to the definition here in il. Some equations of the preceding sections are completely general; others apply only if the two discriminants are polynomials.

If the degree of the differential equation is finite, Ap will be a polynomial in p. On tiie otiier hand, the coefficients of p and its powers are unrestricted so Ac need not be a polynomial.

If either Ap or Ac contain transcendental functions, see 11, the singular solutions may be much more complicated than those which have been discussed here.

Such oases have been treated by Hill. ISie differential equation is of infinite degree since the expansion of P would yield an infinite series in p.

Such equations are not common and they seldom arise in problems of applied mathematics. Some of the preceding metiiods are directly applic- able to equations of this type.

As another possibility, seek a change, iff variable to convert P into p and then use one of the methods of tikis part. Some examples of such differential equations will be found in Part H.

If the second derivatiTe is missing, return to A ; if derivatives occur of order higher than two, refer to C.

It should be noted that the decomposition of a single equation into a S3rstem of equations is not unique.

They are frequently convenient when matrix algebra is to be used, for numerical solutions of equations, and for the study of properties of differential equations.

There are two types of second-order equations. Classify the given equation and proceed as directed. The Hntw-r equation. Consult Bl. The nnnlinaftr equation.

This is the general equation of order two but not of type a; see B 2. There are two main cases ; see a and b. The homogeneous equation.

The word homogeneous is used with a different meaning from that of Al There, it refers to the homogeneous form of the coefficients; see also B Here, it suggests the similarity to a set of simultaneous homogeneous linear algebraic equations.

The equation is also said to be reduced or without second member. Refer to 1 and following sections until a suitable method is found. The nonhomogeneous equation.

The equation is complete, nonhomogeneous, inhomogeneous, not reduced, or with a second member. If the corresponding homogeneous equation can be solved, it is possible, at least in principle, to solve the nonhomogeneous equation.

The general solution of the related homogeneous equation is the comple- mentary ftinciion of the nonhomogeneous equation.

General properties of the linear equation. Some further properties of the equation and its solutions follow. For more details and proofs of them, as well as more exact statements altout the existence theorems, see appropriate references in the Bibliography.

The general solution of the homogeneous equation. Every solution of 2 is contained in 3. The solution of the nonhomogeneous equation.

The solution Y x is the complcTnentary function of 1. Linear independence. Suppose that yi and yz are two special solutions of 2.

They may, or may not, be linearly independent. To test this property, calculate the Wronskian, a determinant of second order.

One function is thus a multiple of the other and the general solution of 2 ihas not yet been obtained. For one method of finding a second linearly independent solution, see Fundamental basis or set of solutions.

The two linearly inde- pendent solutions of 2 are said to form a fundamental set of solutions or a basis. One is of special interest.

The second solution has zero value and unit slope at xq. Constants of integration. Either or both may be zero and, in that ease, the equation is also of type B However, the methods of this section are probably simpler.

If the coefficient of y" is a constant other than unity, divide the other coefficients by that number so that the standard form results.

If the right-hand side of 1 is a constant not equal to zero, or even a function of x, disregard this fact, solve the resulting homqgentoous equa- tion, and then refer to There are several equivalent forms.

As in that case, alternative solutions are often desired. All of the coefficients are constants, return to 1.

The constant zero is permitted for every term except Aq. See 3; the equation may be reducible to case b. None of the preceding cases occur.

It is likely that several methods could be used, possible that one is much easier than the others for a given equalion, and probable that no simple solution exists.

Consider each of the frdlowing before deciding which to try. See B for different types of homogeneity. The word now refers to the definition of Al-8, not that of 1.

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Notwendig immer aktiv.

There are two conclusions, both for Jb 0. For some references where more details can be found, see Milne-2, Scarborough. Agree, Pommesbude Eröffnen impossible alternative procedures are offered. Some further properties of the equation and its solutions follow. The Legendre Transformation.

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To test a differential equation for exactness, see , where methods for finding the primitive are also given.

If the test for exactness fails, it means that common factors have been removed and an integrating factor is needed.

Such factors can alwajrs be found, in principle, but not always easily in practice. When they are known or can be found, the equation can be made exact and integrated; see The Exact Equation.

The equation is exact and also homogeneous, with degree k — 1. Go to 8, for the solution can be found without quadrature.

The equation is exact but not homogeneous. The solution can be found in two equivalent ways.

The quantities xq, yo in the integrals are arbitrary. They are most conveniently chosen to make the integrations easy, frequently as 0 or 1.

The equation is not exact. Proceed to in the hope of finding some function that will make 3 exact so that the methods of this section can be used.

The Integrating Factor. A general theory, based on the properties of continuous transformation groups, shows that an integrating factor can be found, at least in principle, for a properly classified equation.

The solution of the equation can then be completed by quadratures. The use of group theory for finding the integrating factor will not be described here.

For some appropriate references see Cohen, Ince-1, Lie, Page. Since the integrating factor may not always be easy to find, some other procedure may be simpler.

One possibility is a suitable variable trans- formation, converting the given differential equation to one of the types Iff previously considered.

See also 8 and 9 for further suggestions. If the methods of this section are preferred, try to , in turn.

The number of integrating factors. Two integrating factors. An equation for the integrating factor. Unfortunately, the partial differential equation may not be easy to solve and, if one must use it to find an integrating factor, another method might be preferred for solving the given ordinary differential equation.

However, the general solution of the partial differential equation is not needed; any special solution of it will suffice.

In some cases, the partial differential equation becomes an ordinary differential equation and then it may be easy to find an integrating factor.

Some examples are given in The equation is homogeneous. The integration may be completed without a quadrature ; see 8.

Integrating Factor by Inspection. In some cases, an inte- grating factor is obvious from the form of the given differential equation or it is obtainable after a few lucky guesses.

In making such guesses, the following properties may be helpful. If more formal procedures are preferred, go at once to If possible, separate the equation into two or more parts, one of which is exact and the others inexact.

It is only necessary to find an integrating fgictor for the single inexact part. Then, convert F x, y to a function of u.

Examine the differential equation to see if it contains terms like those in Table 1. In more complicated cases, use the methods of a mr b.

Integrating Factors for Special Equations. It is assumed that an integrating factor was not apparent by inspection, as suggested in , and that more formal procedures are wanted.

When the equation is of some special type, the integrating factor can be given at once, as shown in this section.

Alternatively, certain tests can be made on the equation and, if they hold, the integrating factor follows. For this case, see To some extent, the two procedures duplicate each other.

Proceed with this section or go directly to , as desired. There are a number of special cases. Both M and N cannot vanish.

If either vanishes identically, the reciprocal of the other is an integrating factor. TABLE 1. In this case, see 8, the integrating factor is IjM.

The equation is isobaric. This case is a generalization of d. The exponents need not be integers. Here u, v are any functions of x and y.

Thus, this case is more general than d. The equation is linear; see 2. Exchange of variables. Tests for an Integrating Factor.

A quick test will then show whether or not such an integrating factor exists. To proceed in this way. The success of the method depends on the proper choice of u x, y.

A prominent combination of terms in the differential equation might suggest something to be tried. See also Table 1.

The integrating factor depends on only one variable. If the result is that shown in the first column of Table 2, the integrating factor is that given in the second column.

The letters a and k are constants. For a more general case, see b. TABLX 2. When u - yjx, the condition can be generalized; see c.

The integrating factor, see Table 3, is a homogeneous function of x and y of degree zero ; see 8. However, in the exceptional case, the inte- grating factor is unnecessary since the variables are separable.

Advantage can be taken of the fact that k is unspecified for a homogeneous equation. Select two different values of k 0 and 1 might be suitable and find two different integrating factors.

The general solution of the equation follows from b. When the differential equation itself is not homogeneous, the method of this section will work if 7 is satisfied for some value of k.

Homogeneous Equations and Related Types The word homogeneous is used with two different meanings in the study of differential equations.

For the second meaning of homogeneous, see Bl. The test for homoge- neity can usually be made by inspection. In that case, the methods of , , or may be used.

If the equation is homogeneous, but not exact, and an integrating factor is not to be sought, see Some related equations are presented in ff.

The Homogeneous Equation. The given equation has the form 2. The nota- tion means that x is to be replaced in both P and Q by unity and that y is to be replaced by u.

This solution might be contained in 5 or it might be a singular solution; see A Li some cases, polar coordinates are useful.

The Isobaric Equation. It is not necesaaiy that n be an integer. Itisnotneoes- sacy that n be an integer. Take x as the dependent variable and u as the independent variable.

The Jacobi Equation. A generalization of it is given in For a special case, see a. Otherwise, there are three different ways of solving 6 ; see b, c, d.

A special case. The general case. Finally, take X as the dependent variable and a Bern- oulli equation is obtained; see 5.

An alternative method. The Darboux equation, see , is a generali- zation of the Jacobi equation. Methods of solving the former can also be applied to the latter.

Equation numbers here refer to those shown in All Ki are different. A third method. This one, quite different from b or c, depends on solving three simultaneous equations with constant coefficients.

Such equations are not treated in this book but the particular one needed here is quite simple. Details can be found in a number of places, for example, Ince-1, Kaplan- 1.

Its characteristic roots are Ki and its arbitrary constants are Ci. Thus, the solution of the simultaneous system of equations will also give the solution of the Jacobi equation.

There are three cases. When the variable t is eliminated, they will correspond to the three cases of c. All are different.

The Darboux Equation. This is a generalization of , where the fi x, y are polynomials in x and y of maximum degree m and at least one of them is actually of degree m.

The required conditions in the general case were obtained by Darboux. The details may be found in Ince-1 or Goursat.

Here, we only state the two condi- tions which lead to the complete solution of the Darboux equation. The Ki can be found from 8 and the general solution of the differential equation re- sults without quadratures.

Two oases arise. The determinant of the coefficients of the pi vanishes. The case has become equivalent to a. The determinant of the coefficients does not vanish.

Change of Variable When an equation does not fit into one of the previous types, it can often be made to do so by a suitable transformation of variables.

In fact, some of the preceding methods were based on a variable change. It was stated in a that if an equation has a unique solution it will have an infinite number of integrating factors.

Similarly, if it has a unique solu- tion it can be solved by a change of variable. In practice, it may not al- ways be easy to find either an integrating factor or the proper new vari- able.

Sometimes, one method is preferred; at othw times, the seomd method is successful. The following sections contain a few sqggestions that might be helpful if one or two new variables are sought.

Many examples of tiik type will be found in Part II. New Independent Variable. Interchange x and y. The former becomes the new dependent variable and the latter, the new independent variable.

This is an especially simple trick for it requires little calculation and it is easy to see whether the equation is so converted into a known type.

Two examples, where this method was successful, were given; see h and New Dependent Variable.

No general rules can be given, but a conspicuous function in the given equation is often suggestive.

Three rather general equations of no previous type, where this procedure works, are given in b, c, d. Some hints are presented in a.

It is not necessary that m and n be integers in this section. The new variable is u x. Try the following cases.

Other- wise, it might be chosen as some function of x which appears in the original equation. See also 2, , , , , , , , , , , and Part II.

In the first case, try or trigonometric functions like sftiay, cos ay, tan ay, etc. For examples, see , 5, 6, and Part II.

This equation is reminiscent of several types from other sections. Whoi fix a 0, a Bernoulli equation results; see 5. None ofthe preceding special oases occur.

This equation, like that of b, is similar to, but not identical with, several previous types. There are several cases.

It is a Biccati equation; see It is an Abel equation; see Two New Variables. The Legendre Transformation. Also take a new dependent variable, Y — xp —y.

The method of this section is often applicable to equations of degree two or higher; see A2. The solutions would then all be simple combinations of algebraic or elementary transcendental functions; see 3.

Mathematicians, however, have preferred another procedure and have sought algebraic differential equations which define new transcen- dental functions through their solutions.

In doing so, they have studied oertaiR classes of differential equations which will be presented in the apjffopriate places later in this book.

Solutions as an infinite series; see Solution by an approximate method; see In the usual case, a will be preferred.

The remaining parts Of tiiis section summarize some general properties of 1. They are inserted here for convenience of reference and can be consulted as needed.

Analytic Functions. It is the purpose of this section to consider some properties of analytic functions as used in the study of differential equa- tions.

For further details and proofs of the statements which follow, see appropriate references in the Bibliography, for example, Ince-1, Codding- ton and Levinson.

The test for analyticity can be be made simply, for it is only necessary, according to 3 , that the function and its derivatives exist at the point in question.

There are several kinds of singular points. Such a singularity is called a poh of order n or a nonessential singular point.

The misbehavior of the function is effectively avoided by use of the multi- plicative term. If a circle can be drawn with center at the singular point so that no other singular point is enclosed, the singular point is an isolated one.

Within a circle of radius less than 2ln, there are an infinite number of poles. Essential singular points. It is said to be an essential singular point.

Branch points. Whatever the order, some derivatives of finite order and all higher ones will be infinite at a branch point.

Fixed and movable singular points. Examination of the coefficients in a differential equation will reveal the nature of its singular points, which can be of the kinds described in a, b, c.

These are the fixed or intrinsic singular points of the differential equation. It does not follow that solu- tions of the equation will also have singular points of the same kind.

Only under special conditions is the origin a singular point of the solution. Such a singular point, which moves about as the initial values are varied, is a movcMe or parametric singular point.

Singular points for linear equations are always fixed. A nonlinear equation of first order and of first degree can have movable poles and movable branch points but no movable essential singularities.

Nonlinear equations of second or higher order can have movable singular points of all kinds. Movable branch points and essential singularities, excluding poles of finite order, are often called critical points of the differential equation.

There are several cases, depending on the nature of fix, y. Proceed as foUows. Since the latter constant is arbitrary it will be the constant of integration.

For some comments on convergence, see d. The method of undetermined coefficients. Replace the left-hand side of the differential equation by the derivative of 2.

The result, which no longer contains y, is an identity in x. Equate coefficients of equal powers of x in this equation and obtain relations with which Ai, Az,.

There are three possibilities. All coefficients after At are zero so Uiat 2 is a polynomial of degree k. The present procedure can, of course, be used in such cases, if desired.

A general law results for the coefficients in terms of Ao. Such a relation is called a two-term recursion formula or a, first-order difference equation; see also iii.

There is no general law for the coefficients so that three-term for- mulas or even more complicated ones occur.

This means that At will usually depend on At-z, etc. In fact, it may depend in some rather involved way on all of the coefficients which precede it and no explicit solution can be found for A as a function of Aq.

It will become more and more laborious to calculate successive coefficients but, nevertheless, such calculations may be continued as long as desired.

Many-term recursion formulas are linear finite-difference equations. It is often convenient to study them by such methods; see, for example, Jordan, Milne- 1, MOne- Thompson.

Frequently, one wishes a series solution so that y can be determined within some specified limits of error.

This will fix the number of coeffi- cients which must be calculated; see also d. It should be noted that the existence theorems guarantee that the solution is valid but it is not possible to make tests for convergence, as in ii, since the general term is unknown.

Expansion in a Taylor series. The method is equivalent to that of a, which in the usual case will be easier to apply. Use this result, together with and fy, to calculate y" in 3.

Ciontinue in this way to find the third derivative and as many more terms as may be wanted. Evaluation of the successive derivatives can become quite complicated, if it is necessary to use more than a few terms in the series ; see also d.

The coefficients by integration. For an alternative way of deter- mining the coefficients in the series 2 , repeated integrations may be used rather than differentiations.

For the details, see Convergence of the series solution. If the conditions required by the existence theorem hold, it is certain that 2 is the general solution of 1.

A more exact statement of these conditions may be useful. If further details and proofs are wanted, other sources must be consulted.

Some suitable references are Coddington and Levinson, Goursat, Ince Take Jf equal to or greater than this sum, so that! If t is fixed, the right-hand side of 4 will decrease as x decreases.

It is still possible, in this case, to solve the differential equation. All coefficients vanish at X 0, which means that each term in the right-hand member of the differential equation contains x as a factor.

The Equation of Briot and Bouquet. This is case a of For a generalization of it, see There are three possible situations.

Successive values of the At are obtained from 7. Nonanalytic solutions may also exist ; see c. There are relations between k and the coefficients atj so that 7 can be used to calculate the coefficients At.

The preceding result can be obtained in another way. The last differential equatipa is similar to 5 but the coefficient of u has been reduced by unity.

There are no solutions analytic at the origin but there is a general solution con- taining an arbitrary constant. It is a series in x and x In x; it approaches zero as x approaches zero along a properly chosen path.

For references and further details, see Ince Nonanalytic solutions. The first coefficient cqo is arbitrary. The series approaches zero as x approaches zero along a properly chosen path.

More complete details are given by Goursat, Ince-1, Valiron. V The Generalized Equation of Briot and Bouquet. It is mMiiiTiftd that both P and Q are convergent double series in x and y, aimiUr to the right-hand member of 5.

However, P and Q are not divisible by any power of x or y. They are described, for example, by Ince 1. Uo is a simple root. The equation is now of type and can be handled further as described there, b.

Uo is a double root. There are two conclusions, both for Jb 0. The origin is an essential singular point; see lO-I. There are two special cases of interest, which in terms of the original variables, are as follows.

The second equation is of Ricoati type; see Thus 2 is an integral equation which should be solved. Continue in this way and calculate The work is stopped with a solution which is a sufficiently close approxi- mation to the exact solution.

The procedure may become quite tedious, since the integrations are usually more and more difficult to perform.

With appropriate restrictions, existence theorems show that the successive integrals converge and that the solution is unique. It may then be possible to find an approximate solution by one of the following means: a.

For these reasons, we limit the descrip- tion of them here. Graphical methods. Considered geometrically, the differential equation assigns a slope to every point x, y in the x, y-plane.

Each will determine a small portion of an integral curve. The general shape of that curve will become apparent as the plotting is continued.

A smooth curve with the line segments as tangents will be the integral curve sought. This situation is equivalent to the general solution if xq is a constant and yo is a variable parameter or the constant of integration.

Following these general principles, many ingenious variations have been devised. As one possibility, isoclines are first drawn. These are the loci along which the slope required by the differential equation has a constant value.

When a number of isoclines have been drawn, the integral curves can then be sketched on the same plot.

As another possibility, a first approximation to an integral curve could be drawn and this could be improved by redrawing, until two successive curves coincide.

To compute correction. In other cases, a nomograph might be constructed for calculating the corrections.

No high order of precision can be expected by gra]thical means but the results may be acceptable for special problems, especially if the graphs are carefully constructed and of sufficiently large size.

Lacking some par- ticular reason for using this method, one of those in c would usually be preferred.

For more details about the graphical methods, the book of Levy and Baggott is recommended. Mechanical methods. Devices for mechanical integration have a long history.

Two such instruments are the integraph and the polar planimeter. The former, described by Abdank-Abakanowicz in , contains a tracing point, which movps along a graph of the integrand.

An attached pen draws the integral curve. The polar planimeter, one model of which was invented by Amsler in , also has a point, which moves along the integrand curve.

A coimected scale and vernier indicates the number of complete or fractional revolutions made in tracing out the perimeter of the integrand curve.

A simple calibration and calculation give the area under 'this curve, which, of course, equals the value of the definite integral. Either of these devices could be used to solve a simple differential equation in several ways.

For example, the integral equation method, see 12, would give successive approximations to the solution, when the various integrals had been evaluated mechanically.

A wide variety of more sophisticated instruments has been described in the literature and many of them actually constructed.

In some oases, the machine has been invented to solve a particular type of equation, such as that of Bicoati, see 3, or of Abel, see 4.

In other cases, the machine may be more versatile but none of them are simple and all are expensive to build. For many references and further description of such instru- ments, see Kamke.

It was based on addition and integra- tion. The former was achieved by gear boxes and the latter, by a wheel and disk mechanism, similar to that of the polar planimeter.

Originally, a curve was followed manually by the user of the machine. Later, photo- electric curve tracers were added and, eventually, the moving parts were replaced by electronic components.

These developments were a logical outcome of the Mallock electronic machine, which had been invented for solving simultaneous equations.

The modem versions of these machines are called analogue computers. For more details, see Johnson, Soroka.

Numerical methods. The methods of a and b are based on measure- ment and on the properties of smooth curves. The somewhat inexact title of this section suggests that number is now of major importance.

Thus, all operations in these methods are essentially addition or subtrac- tion. They can be carried out by any of the commercially available desk calculating machines or by those high-speed devices known as digital computers.

In general, numerical methods are based on step-by-step integration. The broken-line curve resulting would be a rough approximation to the integral curve desired.

Obviously, the same procedure could be followed with numbers as it is certainly not necessary to draw the curves.

The solution of the differential equation would follow from the Taylor series method of lb. When the derivatives are not easy to obtain, the method could be modified by using interpolation formulas for them.

Alternatively, the integral equations of 12 could be used. If the integration is difficult, use numeri- cal integration.

A slightly different method is that of Bunge and Kutta. In the usual case, it may be the most suitable procedure if a desk machine must be used.

One of these numerical methods may be prefinrred, even though a com- plete solution of the differential equation can be found by anotoer method.

It may then be easier to solve the differential equation numerically than to solve the transcendental equation. Numerical methods are not limited to equations of first order but may be extended to equations of order two or more and to systems of simul- taneous equations.

For some references where more details can be found, see Milne-2, Scarborough. If a digital computer is available for solving the equation, see Wilkes, Wlieeler, and Gill.

If m is a fraction, it will be necessary to rationalize the differential equation and clear it of fractions in order to determine the degree.

However, some of the following methods will apply, even if m is fractional. When transcendental functions of p occur, like Inp or cosp; see Thus, alternative forms of the general solution may be required.

They are described in a, b, c and will be called solutions of types I, II, III, respectively, in the following sections.

In addition to the general solution, a singular solution may also exist. It will satisfy the differential equation but it will not be a special case of the general solution.

When it alone is of interest, go directly to 10, for it may often be obtained without solving the differential equation.

Alternatively, a singular solution may also appear in the methods which follow, when the general solution of the equation is sought. However, the singular solution will usually be lost if common factors are eliminated from both sides of an equation or if they are canceled out in a numerator and a denominator.

When a series solution of the differential equation is desired, or whra the behavior of such a solution is to be investigated in the neighborhood of a singular point, see Al, proceed at once to 7.

Solutions of Type I. It does not need to occur algebraically, however, and in that case, the only re- quirement is that C be arbitrary.

Solutions of Type II. The general solution in the form 1 might be a complicated algebraic or transcendental function. Solutions of Type III.

If the algebra is not too complicated, p could be eliminated between the two simultaneous equations in Ft and and the general solution presented as in a or b.

However, in many cases, such algebraic elimination can be formidable. It is then much simpler, and just as satisfactory, to regard p as a parameter.

To emphasize this meaning for p, it will be replaced by the symbol t. Missing Variables If X, y, or both Ere missing, there are several procedures.

Sometimes, one is easier than the others. For an equation of this type, look at all of the subcases before making a choice of the method to be used.

If a new variable is suggested, see Conversion of the equation into type is sometimes helpful; see Solve for p.

Solve for x. There are two possibili- ties. Use a New Variable. Exchange Variables. The new dependent variable will be X and Ijp s dxidy.

It can also be givrni as y x, y, 0 0, which is type I. Solve for y. Replace pbyt; use 6 and 6 as a parametric solution of type III. Both Variables Missing.

Factor it, if possible, to get P -ri p -ra The individual factors can be integrated since the variables are separated in each.

Occasionally, the method of might be useful. Consult 3, 4, 5. Sometunes, the Legoidre transformation, see A1 , will be helpful.

The result is p -Fi p -Fa Since each factor is of first degree, the methods of A1 can be used. Some further information on equations of this case will be found in 8.

Differentiate with respect to y. A first-order equation in y and p is obtained but x is missing. Retain both 1 and 2 as a parametric solution in terms of the parameter t, which has replaced p.

Thus, the solution is type III. Regard y as a function of p. Solve the first-order equation by a method of Al.

Eliminate p between 1 and 3 to get y - -l- C. Differentiate with respect to x. Gall p a variable parameter, rename it t, and give the solution as type III, with both 4 and 6.

Calculate dx Fp dp p -Fg where the subscripts mean partial derivatives; see b. Special cases. If the given differential equation is linear in y, with a constant for its coefficient, the differentiation in a or b can be carried out at once; it is not necessary to solve for y.

Refer to 4. Equations of the special type described there can be solved very easily. Homogeneous Equations and Related Types For the meaning of the word homogeneous, as used here, see Al There are several different kinds of equations.

Proceed in one of the following ways, whichever seems the easiest. Solve fory. Use As another possibility, replace p by t and give a parametric solution of type III.

Introduce a New Variable. The form of the equation may suggest the appropriate transformation. Then, whichever seems easier: i.

Solve for x and proceed according to b. If algebraic difficulties are severe, replace v by the parameter t and give a solution of type III.

The method of is applicable so that. J Restore the original variables or give a parametric solution as in a. It is type and its solution can be found by a method of that section.

Glairauf 8 Equation and Related Types There are three dififerent equations, each quite similar in form.

Glalraut's Equation. It contains no arbit- rary constant; it is not a special case of the general solution 1 ; it satisfies the differential equation.

Alternatively, 1 and 4 may be used as a para- metric solution of type III. In either case, the result is a singular solution ; see 10 for further details.

If y p R 0, the equation is homogeneous; see Make x the dependent variable and p the indepmident variable.

Alternatively, retain 5 and 7 as a solution of type III, replacing p by a parameter t. A singular solution may also exist; see The Legendre transformation.

Alternatively, differentiate 1 with respect to x, solve the resulting second-order equation, see B, to get 2 and use 3 to eliminate one constant.

Change of Variable. Try to convert the differential equation into one of the preceding types by a new dependent variable, a new independent variable, or two new variables.

Specific directions are not readily given but a number of suggestions may result from Al Modifications needed for equations of second or higher degree are usually obvious.

Since an equation of Clairaut t 3 pe, see , is quite simple to solve, seek a transformed equation of that kind, as one possibility.

A few special cases of variable transformation are listed in the next section. Many further examples will be found in Part II. In that case, it may be possible to use: a.

An infinite series; see Al A definite integral; see Al- An approximate method; see Al The refierences apply to an equation of the first degree but they may be modified for equations of higher degree.

Since problems of this sort are not common, no further details will be given here. As stated in Al, mathematicians have been interested in finding certain special classes of differential equations which define new tnuosoen- dental functions.

The necessary conditions for equations of degree tvro or greater are stated in 8 and its subsections. With still further restrictions, the special cases of 9 result.

Note that the subscript in 3 does not mean a partial derivative, as it does in 2. There are four cases. Ap vanishes independently of y.

Xo vanishes independently of y. There are singular points of Xi for general values of y. There are singular points for a root of 3.

Exclude each of them from subsequent consideration, for we are interested here only in the movable singular points. Let zo, ya be some point other than those which have been excluded.

It may, or may not, be a movable singular point. Series solutions of 1 are wanted in the neighborhood of zq, yo. There are four special cases.

A study of them will reveal the conditions fox a solution with no movable singular points. If that case along is of intexest, go to 9.

Each is an analytic function there. Treat each of the other roots in the same way. The general solution of 1 will be, see Use to find m-1 solutions for the finite roots.

For general values of y there will be ih different roots of p; see Suppose that the roots equal to 9 are pi, p 2 ,. Then, one of the equal roots, pi for example, will either return to its initial value or become equal to one of its partners ps, P8,.

There are several special oases. The differential equation is analytic in x and Y. Its solution is an analytic function; see Al Three possibilities arise in this case.

It is a singular solution; see This solution has a movable branch point. This is the same situation as that in the first case of ii.

The conclusions are similar to those in , except that the multiple root is at infinity. X, y pm-i Other special cases of 8 are presented in 9- Iff.

For further details about the equation, see the following parts of this section. Movable branch points. To avoid them, require that JCq be a function of X alone.

A singular solution. A restriction on a. Further restrictions. Suppose also that P is a many valued function of u.

It is required that tt be a factor of both fm and fm-u t. A branch of order a. It is necessary, according to c, that a - 1 C k.

No movable essential. This is true for 1 , independ- dent of the presence or absence of movable branch points.

There can be no fixed singular points, except possibly one at infinity, since the coefficients do not depend on x.

If all conditions of 9 are met, there can be no movable branch points. Furthermore, there are no essential singularities for finite X.

The point at infinity can be an essential singular point, but not a branch point. This is a special case of the general equation in 9.

It is assumed to be irreducible, see 8, and to satisfy all further re- quirements of 9. If uninterested in the details, go to , where these permitted oases are listed.

If the arguments lor 3 are wanted, see the following. If the degree is less than 2m, there are two possibilities, but both can be reduced to the case where the degree is 2m.

Por convenience, the degree of Xlx, y will be taken as exactly 2m. The degree of Xix, y is less than 2m, but it does not contain y as a factor.

It has become of degree 2m in the variables u and x. The degree of X x, y is less than 2m, but it contains y as a factor.

There are equal roots. This means that Ap a;, y s 0; see and When the differential equation of is restricted so that there are no movable branch points, the equation of this section results.

The results are listed in a. There is only one possible differential equation; see b. The Binomial Equations. If all conditions in are met, the resulting equations are called binomial.

For a method of solving each differential equation, see Part II. The oases of a. There are six possible types ; see Table 1.

It becomes type II, if is a constant, not equal to 02 or Degenerate cases. Each will give one or more degenerate cases of degree lower than 2m.

The possible cases are listed in Table 2. TABLE 2. It satisfies the differential equation. It contains no arbitrary constant.

It cannot be obtained by assigning a particular value to the integra- tion constant in the general solution of 1.

Such a function is a aingvJar aolvJtion. Its properties are frequently of interest and, in some problems, it is wanted rather than the general solution of the differential equation.

The equation is reducible. There will be no singular solution. A check on this situation can be made, for the p-discriminant, see , will vanish identically.

The equation is irreducible. A singular solution may exist. In such a case either or could be used.

The former may be simpler, since the general solution of the differential equation need not be known. However, it would be preferable to complete the work of both and , in that order.

Then, refer to for a more complete treatment of the problem. Juni 3, admin. Inhalt 1 icd 10 asthma bronchiale 2 j45 1 g 3 asthma bronchiale intrinsicum et extrinsicum 4 j This website uses cookies to improve your experience.

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