Chapter 4.7, Problem 24P

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.

$\left(1-t\right)y^{″}+ty\text{'}-y=2{\left(t-1\right)}^2{e}^{-t},0<t<1$;      $y_1\left(t\right)=e^t$, $y_2\left(t\right)=t$