Differential Equations, 3rd Edition

13Qualitative Properties and Theoretical Aspects

DOI: 10.1201/9781003214526-13

• The general solution

• Existence and uniqueness theorems

• The Wronskian

• Linear independence

• The Sturm theorems

13.1 A Bit of Theory

Until now, in our study of second-order and higher-order differential equations, we have spoken of “independent solutions” without saying exactly what we mean. We now take the time to discuss this matter carefully. The reader may find it useful to refer to the appendix on linear algebra in order to put these ideas in context.

A collection of functions ${\varphi }_1,\dots ,{\varphi }_k$ defined on an interval [a, b] is called linearly independent if there do not exist constants $a_1,\dots ,a_k$ (not all zero) such that

$$\begin{array}{l}{a}_1{\varphi }_1+\cdots +a_k{\varphi }_k\equiv 0.\hfill \end{array}$$

If the functions are not linearly independent ...