Chapter 8.6, Problem 27P

Verify that (6.4) is a particular solution of $(6.2).$ Verify that another particular solution of (6.2) is $y_p=\frac{1}{10}\sin 2x-e^{-x}$.

Observe that we obtain the same general solution (6.7) whichever particular solution we use [since ( $A-1$ ) is just as good an arbitrary constant as $A$ ]. Show in general that the difference between two particular solutions of $\left(a_2{D}^2+a_{1}D+a_0\right)y=f(x)$ is always a solution of the homogeneous equation $\left(a_2{D}^2+a_{1}D+a_0\right)y=0$, and thus show that the general solution is the same for all choices of a particular solution.