Chapter 4.7, Problem 22P

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.

$t^2{y}^{″}-t\left(t+2\right)y\text{'}+\left(t+2\right)y=2t^3,t>0$;      $y_1\left(t\right)=\mathrm{t}$, $y_2\left(t\right)=\mathrm{t}{e}^t$