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.
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 .
Two solutions and are called a fundamental pair if any other solution is expressible as
where and are constants. A fundamental pair can be obtained, for example, by taking any and requiring that
The Wronskian of and is defined by
(More generally , where .) Then the following relation is known as Abel’s identity
where is independent of and is defined in (1.13.1). (More generally in (1.13.5) for th-order differential equations, is the coefficient multiplying the th-order derivative of the solution divided by the coefficient multiplying the th-order derivative of the solution, see Ince (1926, §5.2).) 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
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 .
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
where and are constants.
The substitution in (1.13.1) gives
in (1.13.1) gives
In (1.13.1) substitute
Let satisfy (1.13.14), be any thrice-differentiable function of , and
Here dots denote differentiations with respect to , and is the Schwarzian derivative:
For arbitrary and ,
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).
For an extensive collection of solutions of differential equations of the first, second, and higher orders see Kamke (1977).