Chapter 1.3, Problem 33P

Verify that the function $\varphi \left(t\right)=c_1{e}^t+c_{2}t{e}^t$ is a solution of the linear equation $y\text{'}\text{'}-2y\text{'}+y=0$ for any choice of constants $c_1$ and $c_2$. Determine $c_1$ and $c_2$ so that each of the following initial conditions is satisfied:

$y\left(0\right)=3,y\text{'}\left(0\right)=1$

$y\left(0\right)=1,y\text{'}\left(0\right)=-4$