Chapter 4.7, Problem 23P

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.

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