Verify that the given functions y 1and y2 satisfy the corresponding homogeneous equation, then find a particular solution of the given nonhomogeneous equation.
$$
t^{2} y^{\prime \prime}-t(t+2) y^{\prime}+(t+2) y=2 t^{3}, t>0 ; y_{1}(t)=t, y_{2}(t)=t e^{t}
$$
$$
y_{1}(t)=t \quad y_{1}^{\prime}(t)=1 \quad y_{1}^{\prime \prime}(t)=0
$$
$$
-t(t+2)+(t+2) t=0
$$
$$
y_{2}(t)=t e^{t} \quad y_{2}^{\prime}(t)=e^{t}+t e^{t} \quad y_{2}^{\prime \prime}(t)=2 e^{t}+t e^{t}
$$
$$
t^{2}\left(2 e^{t}+t e^{t}\right)-t(t+2)\left(e^{t}+t e^{t}\right)+(t+2) t e^{t}=0
$$
Hence, $y_{1}$ and $y_{2}$ satisfy the homogeneons equation
$$
y^{\prime \prime}-\frac{t+2}{t} y^{\prime}+\frac{t+2}{t^{2}} y=2 t
$$
$$
\begin{aligned}
w&=\left|\begin{array}{cc}{t} & {t e^{t}} \\ {1} & {e^{t}+\operatorname{te}^{t}}\end{array}\right|=t^{2} e^{t}\\
w_{1}&=\left|\begin{array}{cc}{0} & {t e^{t}} \\ {1} & {e^{t}+t e^{t}}\end{array}\right|=-t e^{t}\\
w_{2}&=\left|\begin{array}{cc}{t} & {0} \\ {1} & {1}\end{array}\right|=t
\end{aligned}
$$
$$
\begin{aligned} y_{p}(t) &=t \int \frac{\left(-t e^{t}\right)(2 t)}{t^{2} e^{t}} d t+t e^{t} \int \frac{(t)(2 t)}{t^{2} e^{t}} d t \\ &=t \int-2 d t+2 t e^{t} \int \frac{1}{e^{t}} d t \\ &=t(-2 t)+2 t e^{t}\left(-e^{-t}\right) \\ &=-2 t^{2}-2 t \end{aligned}
$$
