Chapter 4.7, Problem 26P

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.

$\left(1-x\right)y^{″}+xy\text{'}-y=g\left(x\right),0<x<1$; $y_1(x)=e^x$, $y_2(x)=x$