Chapter 4.7, Problem 25P

In each of Problems 22 through 27, verify that the given functions $y_1$ and $y_2$ satisfy the corresponding homogeneous equation; then find a particular solution of the given nonhomogeneous equation. In Problems 26 and 27, $g$ is an arbitrary continuous function.

$x^2{y}^{″}+xy\text{'}+\left(x^2-0.25\right)y=3x^{3/2}\sin x,x>0$; $y_1(x)=x^{-1/2}\sin x$, $y_2(x)=x^{-1/2}\cos x$