Solving Second-Order Differential Equations by Decomposition

1. Introduction

The history of differential equations begins shortly after the establishment of the analysis by Newton and Leibniz in the 17th century. A brief overview of its first hundred years can be found in Appendix A of Ince’s book [1]. These early investigations were mainly limited to first-order equations, associated with the names Riccati, Bernoulli and Euler. Starting in the early 18th century special linear equations of higher order were also investigated.

A more systematic search for solution methods was initiated by the results of Galois for solving algebraic equations in the early 19th century. Inspired by these results, Picard and Pessiot in Paris founded a solution theory for linear differential equations, known as Picard-Vessiot theory or differential Galois theory. A good introduction into their work and its extensions by Loewy may be found in the books [2, 3]. Completely independent of these activities Sophus Lie in Leipzig founded the so-called symmetry analysis for solving nonlinear differential Equations [4, 5]. Its main weaknesses are that most differential equations have no symmetries and therefore it cannot be applied. Furthermore, there are many differential equations with fairly simple closed form solutions that have no symmetries. That was essentially the status in the early twentieth century, which did not fundamentally change until its end.

In this situation, a new solution method based on decompositions was proposed [6]. Essentially a decomposition means to find a component of lower order such that the original equation may be represented as a differential polynomial in terms of this component. Its existence is based on the following observation. Let Fxyy′y″=0be a second-order differential equation for a function ydepending on a variable x, and ωxyC1C2=0its general solution depending on two undetermined constants. It describes a two-parameter family of curves in the x−y-plane. If C1and C2are constrained by a relation φC1C2=0the resulting expression for ωcontains effectively a single parameter C. It describes a family of curves that may obey a first-order differential equation called a component. Its solutions are also solutions of the originally given second-order equation.

Every second-order equation has an infinite number of first-order components corresponding to the choice of φC1C2. Any such component has the form

Its meaning may be described as follows. If z≡gxyy′Cis substituted into fxyzz′Cthe second-order equation on the left-hand side is obtained. The constant Cdoes not necessarily occur in fand g, the same is true for yand its occurence in f.

Solving a second-order equation by decomposition involves two steps. First a decomposition of a certain type has to be found. Then the first order equation has to be solved in order to get the solutions of the original second-order equation. Of particular interest are those components the solution of which can always be determined. These are linear homogeneous and inhomogeneous components, special Riccati components, Bernoulli, Clairaut or d’Alembert components.

In this article equations of second order for an unknown function ydepending on xwith leading term y″or y′y″are considered. They are assumed to be linear in y″, polynomial in the derivatives y′, and rational in yand x. Equations of this kind are fairly common in applications, therefore many special examples of them are given in the collections by Kamke [7], Murphy [8], Polyanin [9], Sachdev [10] and Zwillinger [11]. Many interesting applications of such differential equations can be found in the textbooks by MacCluer et al. [12] and Swift and Wirkus [13].

In the following Section 2 equations with leading term y″are considered, and possible linear or Bernoulli components are determined. For linear inhomogeneous equations it is shown how decomposition leads to a new procedure for determining a special solution without first having to know a fundamental system. Equations with leading term y′y″and possible components of Clairaut or d’Alembert type are the subject of Section 3. Most of the examples do not have Lie symmetries, so decomposition is the only way to solve them. The last Section 4 discusses various possible generalizations of the decomposition method, on the one hand more general equations to be solved, on the other hand more general first-order components.

2. Equations with leading term y″

Equations that are linear in the highest derivative y″, but may contain powers of y′with coefficients that are rational in yand xare considered in this section. Moreover it is assumed that they are primitive, i.e. the leading coefficient is unity. Their general form is

Equations of this form appear in numerous applications, as can be seen in the collections of solved examples quoted above. The following proposition has been proved in [6], it is the basis for generating quasilinear first-order components; as usual y′≡dydxand D≡ddx.

Proposition 1 Let a second-order quasilinear Eq. (2) be given. A first-order component z≡y′+rxyexists if rxysatisfies

Then the original second-order equation can be decomposed as

The proof may be found in Section 2 of [6]. As a first application linear first-order components of the form z≡y′+axy+bxare searched for, i.e. with the above notation rxy=axy+bx; its coefficients aand bare solutions of the so-called determining system, they may be in any field extension of Qx. The following proposition describes how they may be obtained.

Proposition 2. Let a second-order quasilinear Eq. (2) be given. In order that it has a linear first-order component z≡y′+axy+bxthe coefficients axand bxhave to satisfy

Then (2) may be written as follows

The coefficients aand bare solutions of a first-order algebro-differential system. Its general form is

for i=1,2…; pabx, qabxand riabxare polynomials in aand b, and rational in x; their maximal degree in aand bis K. The riabxgenerate an ideal Iab∈Qxab.

Proof. Substituting r=axy+bxinto (3) yields (5). At this point yis considered as an undetermined function. Therefore the left-hand side of (5) is represented as a partial fraction in y. Equating its coefficients to zero yields sufficient conditions in order that (5) vanishes and zis a component of (2). The first order ode’s for aand bin the determining system (7) originate from the coefficients of first and zeroth degree in yof (5). The polynomials in aand b, i.e. pabx, qabxand riabxin (7) originate from the powers of ay+band the rational coefficients ckxyin (5), i.e. exclusively from the nonlinearities of (2). Substitution of y′=z−ay−binto (4) yields (6). As a result, the sums at the left-hand side of (6) are a polynomial in zthe coefficients of which may depend explicitly on y.

It is important to represent the left side of (5) as a partial fraction in y, only in this way the structure of the system (7) is assured.

3. Equations with leading term y′y″

Another important class of differential equations are those with leading term y′y″, they are considered in this section. Their general form is

Components of Clairaut or d’Alembert type z≡y−xfy′−gy′may lead to partial or even general solutions in closed form, mostly in a parameter representation. The main result of this section is given in the following proposition.

Proposition 4 Let a second-order differential Eq. (24) be given. A first-order component z≡y−xfy′−gy′exists if fy′and gy′satisfy

where p≡y′has been defined. Representing the left hand side of (25) as a partial fraction w.r.t. xand equating the coefficients of the various terms to zero, a system of first-order quasilinear ode’s for fpand gpis obtained; its degree in fpand gpis not higher than the degree in yof the coefficients cxyand ckxy.

Proof. Reduction of (24) w.r.t. z≡y−xfy′−gy′leads to Eq. (25). Their properties follow directly from the assumptions about the coefficients cxyand ckxyin (24), and representing the left hand side of (25) as a partial fraction in x.

The determining system for the two functions fpand gpmay be obtained explicitly from (25) if the coefficients cxyand ckxyare known. Without restrictions on the coefficients ckxythe derivatives f′pand g′pmay occur linearly in any equation obtained after separation w.r.t. x, and an algebraic system in p, fp, gp, f′pand g′pfollows. It turns out that an algebraic Gröbner basis algorithm including factorization is a suitable tool for solving them in many cases. If a solution has been obtained the corresponding component may be applied for generating the decomposition of the given equation explicitly. The following example uses this proceeding.

Example 10 Consider the equation

Here J=1and K=2, its nonvanishing coefficients are cxy=12y, c2=x−12x, c1=−y2xand c0=12x. A linear or Bernoulli component does not exist. Proposition 4 leads to the system

Transforming the left-hand sides into algebraic Gröbner bases in the term order f′p>g′p>fp>gp>p, the following two systems and their solutions are obtained.

They lead to the decompositions

respectively. The former decomposition generates a Clairaut component. It yields the solution y=Cx+1C−Cof (26), Cis an undetermined constant. Its parameter solution x=1p2+1, y=2pdoes not solve it, it annihilates a lower-order factor of the expression in the left-hand bracket of this decomposition and has to be discarded.

Integration of the d’Alembert component z≡y′2+yy′−Cx−1leads to the general solution of (26) in a parameter representation

Eq. (26) does not have a Lie symmetry. □

This example shows that solutions of a component must be tested to see if they meet the second order equation, otherwise they have to be discarded; this phenomenon seems to be quite common.

4. Conclusions

The structure of the determining systems for linear or Bernoulli components of a nonlinear Eq. (2) given in Propositions 2 or 3, respectively, show clearly its relation to the corresponding system for the decomposition of a linear equation. For a generic equation of the second order this appears to be the best possible result. The same applies to the verious solution steps given on page 6. The corresponding result for determining Clairaut and d’Alembert components given in Proposition 4 is less specific. However, it should be possible, to obtain more detailed results if special classes of second-order equations are considered. In general, this area is only at an early stage and a better understanding of the underlying mechanisms generating the solutions and also its limitations would be highly desirable.

There are numerous possible generalizations fairly obvious. On the one hand, this concerns the equations to be solved. More general function fields for its coefficients like e.g. algebraic or elementary functions may be allowed. Equations of order three or four would be interesting in many applications. The greatest challenge however is certainly to develop similar procedures for partial differential equations as it has been indicated in Section 5 of [6].

On the other hand, the component type offers space for extensions too. In principle all equations of first order, as described for example in Kamke’s book [7], Part A, Section 4, are possible components. Components that guarantee at least a partial solution are of course particularly useful, the most important of them have been discussed in this article.

In order to apply decompositions to concrete problems the implementation of the procedures described in this article are available on the websitewww.alltypes.de[15].

Beyond that there are a number of general problems related to decompositions. For instance the question how rare are equations that allow a particular decomposition, Example 6 provides a partial answer. If two or more one-parameter families of solution curves are known as in Example 4, does it faciliate generating the general solution? The exact relation between Lie’s symmetry analysis and solution by decompositions is another subject of interest.

Acknowledgments

Various helpful comments of a referee are gratefully acknowledged.