Chapter 2.4, Problem 26P

Show that the solution (7) of the general linear equation (1) can be written in the form

$y=c{y}_1\left(t\right)+y_2\left(t\right)$, (i)

Where $c$ is an arbitrary constant. Identify the functions $y_1$ and $y_2$.

Show that $y_1$ is a solution of the differential equation

$y\text{'}+p\left(t\right)y=0$, (ii)

Corresponding to $g\left(t\right)=0$.

Show that $y_2$ is a solution of the full linear equation (1). We see later (e.g. in Section $4.5$) that solutions of higher order linear equations have a pattern similar to Eq. (i).

Solution (7): $y=\frac{1}{\mu \left(t\right)}\left[{\displaystyle \underset{t_0}{\overset{t}{\int }}\mu \left(s\right)g\left(s\right)ds+c}\right]$.

General linear equation (1): $y\text{'}+p\left(t\right)y=g\left(t\right)$.