More on Second Order Equations

In this section, we wish to study some more properties of second order equations which have nice applications. They also have natural generalisations to higher order equations.

DEFINITION 8.2.1 (General Solution)   Let $y_1$ and $y_2$ be a fundamental system of solutions for

$$y^{\prime\prime} + q(x) y^\prime + r(x) y = 0, \; x \in I.$$ (8.2.1)

The general solution $y$ of Equation (8.2.1) is defined by

$$y = c_1 y_1 + c_2 y_2 , \; x \in I$$

where $c_1$ and $c_2$ are arbitrary real constants. Note that $y$ is also a solution of Equation (8.2.1).

In other words, the general solution of Equation (8.2.1) is a $2$ -parameter family of solutions, the parameters being $c_1$ and $c_2.$