## Abstract

The subject of this article are linear and quasilinear differential equations of second order that may be decomposed into a first-order component with guaranteed solution procedure for obtaining closed-form solutions. These are homogeneous or inhomogeneous linear components, special Riccati components, Bernoulli, Clairaut or d’Alembert components. Procedures are described how they may be determined and how solutions of the originally given second order equation may be obtained from them. This makes it possible to solve new classes of differential equations and opens up a new area of research. Applying decomposition to linear inhomogeneous equations a simple procedure for determining a special solution follows. It is not based on the method of variation of constants of Lagrange, and consequently does not require the knowledge of a fundamental system. Algorithms based on these results are implemented in the computer algebra system ALLTYPES which is available on the website www.alltypes.de.

### Keywords

- ordinary differential equations
- decomposition
- exact solutions
- computer algebra

## 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

Every second-order equation has an infinite number of first-order components corresponding to the choice of

Its meaning may be described as follows. If

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

In the following Section 2 equations with leading term

## 2. Equations with leading term y ″

Equations that are linear in the highest derivative

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

Proposition 1 Let a second-order quasilinear Eq. (2) be given. A first-order component

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 *determining system*, they may be in any field extension of

Proposition 2. Let a second-order quasilinear Eq. (2) be given. In order that it has a linear first-order component

Then (2) may be written as follows

The coefficients

for

*Proof.* Substituting

It is important to represent the left side of (5) as a partial fraction in

### 2.1 Linear equations

If

Corollary 1 Let

be given. A first-order component

If it is satisfied Eq. (8) may be written as

*Proof.* The system (9) follows from (5) for the given special values of

It is remarkable that in the case of linear equations the algebraic conditions

For linear *homogeneous* ode’s, i.e. for

For linear *inhomogeneous* equations in addition to a fundamental system a special solution has to be found. The above corollary avoids the usual method of variation of constants that somehow appears like an *ad hoc* method. The method described in the above corollary requires only a special solution of a Riccati equation and subsequently solving a linear first-order equation in order to obtain the general solution of the second-order Eq. (8). The following example applies this procedure.

Example 1 The equation

is Equation 2.109 in Kamke’s collection [7]. Here

Integration yields the general solution

This is also the general solution of Eq. (11).

It may occur that a fundamental system of a second-order equation is rather complicated. Usually this is the case when the Riccati equation for

Example 2 Consider the equation

Here

Its special solution leads to the component

One more integration yields a special solution of (12).

The application of Corollary 1 is particularly convenient if the coefficients

Example 3 The equation

Its general solution

is also the general solution of the given second-order equation.□

### 2.2 Quasilinear equations

The most interesting applications of Proposition 2 relate to nonlinear equations, of course. They differ from the linear case mainly by the occurence of the ideal

At first the algebraic system

If it is inconsistent a linear component does not exist in any field extension. This applies to a generic nonlinear equation of the form (2).

If the ideal

Finally, the algebraic equations may generate a relation between

Subsequently this proceeding will be illustrated by several examples. They show that all of the alternatives mentioned actually exist.

Example 4 Consider the equation

Its coefficients

The single algebraic equation has the solutions

follow. Integration of the two components yields the two one-parameter families

The most interesting, of course, are equations that allow a one-parameter family of linear components and whose integration gives the general solution. The next example is of this type.

Example 5 Consider the equation

with

with solution

is obtained. Integration of the first-order component leads to the general solution

of Eq. (13), it does not have a Lie symmetry.

Here the question arises how exceptional are the equations that have a one-parameter family of linear components of the first order and thus have a general solution in closed form. The following example is a generalization of the previous one. A family of second order equations is constructed whose general solution can be given explicitly.

Example 6 The equation

with undetermined coefficients

The result may be described as follows. If

exists. Defining

This is the general solution of Eq. (14). □

It turns out that a behavior similar to that in the previous example often applies, i.e. first-order linear components often exist not only for isolated equations, but for entire families, which are parameterized by indefinite functions. This explains the existence of families of solvable equations as those given in the collections mentioned above.

Bernoulli equations are another class of first-order ode’s with guaranteed closed form general solutions. In addition to a term linear in

Proposition 3 Let a second-order quasilinear Eq. (2) be given. In order that it has a first-order Bernoulli component

Then (2) may be written as follows

The coefficients

*Proof.* Substituting

Substitution of

The structure of the system (17) is similar as for linear components considered above, and consequently also the proceeding for its solution. The following examples applies the above proposition.

Example 7 The equation

with

follows. The two equations

This system has the solution

follows. Integrating the right component yields the general solution

of Eq. (18). It does not have a Lie symmetry. □

The next example deals with a problem in hydrodynamics. The boundary layer at a circular cylinder immersed in the uniform flow of liquid is considered [14], see also Eq. 6.210 of [7].

Example 8 The equation

has the only nonvanishing coefficient

It turns out that for

After some simplifications the resulting system for

This is also the general solution of Eq. (20) □

In general it is *a priori* not known whether there exists a Bernoulli component of any order. If a component for small values of

Example 9 Consider the equation

with

Expanding the last term into unique partial fractions by using the general formula

leads to

The coefficients of the various terms yield a system for the unknowns

At the moment an algorithm for determining bounds for

## 3. Equations with leading term y ′ y ″

Another important class of differential equations are those with leading term

Components of Clairaut or d’Alembert type

Proposition 4 Let a second-order differential Eq. (24) be given. A first-order component

where

*Proof.* Reduction of (24) w.r.t.

The determining system for the two functions

Example 10 Consider the equation

Here

Transforming the left-hand sides into algebraic Gröbner bases in the term order

They lead to the decompositions

respectively. The former decomposition generates a Clairaut component. It yields the solution

Integration of the d’Alembert component

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 website

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.