Chapter 6.4, Problem 14E

To determine

(a)

To show:

The general solution to the given differential equation can be written in the form

$\begin{array}{rll}y\left(x\right)& =& C_1+C_{2}x+C_3{e}^{kx}+C_4{e}^{-kx}+\frac{1}{k^2}{\displaystyle \int q\left(x\right)xdx}-\frac{x}{k^2}{\displaystyle \int q\left(x\right)dx}+\frac{e^{kx}}{2k^3}{\displaystyle \int q\left(x\right)e^{-kx}dx}-\frac{e^{-kx}}{2k^3}{\displaystyle \int q\left(x\right)e^{kx}dx}\end{array}$

To determine

(b)

To show:

The general solution in part (a) can be rewritten in the given form.

To determine

(c)

To compute:

The general solution using the formula in part (a) and then using the formula in part (b) and compare these two general solutions with the given general solution.