# §1.13 Differential Equations

## §1.13(i) Existence of Solutions

A domain in the complex plane is simply-connected if it has no “holes”; more precisely, if its complement in the extended plane is connected.

The equation

where , a simply-connected domain, and , are analytic in , has an infinite number of analytic solutions in . A solution becomes unique, for example, when and are prescribed at a point in .

### ¶ Fundamental Pair

Two solutions and are called a fundamental pair if any other solution is expressible as

1.13.2

where and are constants. A fundamental pair can be obtained, for example, by taking any and requiring that

1.13.3

### ¶ Wronskian

The Wronskian of and is defined by

1.13.4

Then

where is independent of . If , then the Wronskian is constant.

The following three statements are equivalent: and comprise a fundamental pair in ; does not vanish in ; and are linearly independent, that is, the only constants and such that

are .

## §1.13(ii) Equations with a Parameter

Assume that in the equation

and belong to domains and respectively, the coefficients and are continuous functions of both variables, and for each fixed (fixed ) the two functions are analytic in (in ). Suppose also that at (a fixed) , and are analytic functions of . Then at each , , and are analytic functions of .

## §1.13(iii) Inhomogeneous Equations

The inhomogeneous (or nonhomogeneous) equation

with , , and analytic in has infinitely many analytic solutions in . If is any one solution, and , are a fundamental pair of solutions of the corresponding homogeneous equation (1.13.1), then every solution of (1.13.8) can be expressed as

1.13.9

where and are constants.

### ¶ Variation of Parameters

With the notation of (1.13.8) and (1.13.9)

1.13.10

## §1.13(iv) Change of Variables

### ¶ Transformation of the Point at Infinity

The substitution in (1.13.1) gives

where

1.13.12

### ¶ Elimination of First Derivative by Change of Dependent Variable

The substitution

in (1.13.1) gives

where

1.13.15

### ¶ Liouville Transformation

Let satisfy (1.13.14), be any thrice-differentiable function of , and

Then

Here dots denote differentiations with respect to , and is the Schwarzian derivative:

1.13.20

## §1.13(v) Products of Solutions

The product of any two solutions of (1.13.1) satisfies

If and are respectively solutions of

then is a solution of

For extensions of these results to linear homogeneous differential equations of arbitrary order see Spigler (1984).

## §1.13(vi) Singularities

For classification of singularities of (1.13.1) and expansions of solutions in the neighborhoods of singularities, see §2.7.

## §1.13(vii) Closed-Form Solutions

For an extensive collection of solutions of differential equations of the first, second, and higher orders see Kamke (1977).