Chapter 4.7, Problem 36E

Verify that the given functions $y_1$ and $y_2$ are linearly independent solutions of the following differential equation and find the solution that satisfies the given initial conditions.

$t{y}^{\prime \prime }-\left(t+2\right)y^{\prime }+2y=0$;

$y_1\left(t\right)=e^t$, $y_2\left(t\right)=t^2+2t+2$;

$y\left(1\right)=0$, $y^{\prime }\left(1\right)=1$