Verify that the given functions of y1 and y2 satisfy the corresponding homogeneous equation; then find a particular solution of the given nonhomogeneous equation.
$(1-t)y''+ty'-y=2(t-1)^{2}e^{-t},0<t<1$
$y_{1}(t)=e^{t}, y_{2}(t)=t$
Hence,
$$\begin{cases}
y_{1}(t)=e^t\\
y_{1}'(t)=e^t\\
y_{1}''(t)=e^t\\
\end{cases} , \begin{cases}
y_{2}(t)=t\\
y_{2}'(t)=1\\
y_{2}''(t)=0\\
\end{cases}
$$
Subsitute back into the homogeneous equation:
$$(1-t)y''+ty'-y=0$$
\begin{equation}
(1-t)e^t+te^t-e^t=0
\end{equation}
\begin{equation}
e^t(1-t+t-1)=0
\end{equation}
\begin{equation}
0=0
\end{equation}
Verified $y_1$ satisfy the corresponding homogeneous equation.
\begin{equation}
(1-t)0+t*1-t=0
\end{equation}
\begin{equation}
t-t=0
\end{equation}
\begin{equation}
0=0
\end{equation}
Verified $y_2$ satisfy the corresponding homogeneous equation.
Now we find particular solution of the given nonhomogeneous equation by first dividing it by $1-t$:
\begin{equation}
y''+\frac{t}{1-t}y'-\frac{1}{1-t}y=2(1-t)e^{-t}
\end{equation}
Then
\begin{equation}
g(t)=2(1-t)e^{-t}
\end{equation}
\begin{equation}
W[y_1,y_2]=\begin{vmatrix}
{y_{1}(t)}&{y_{2}(t)}\\
{y_{1}'(t)}&{y_{2}'(t)}\\
\end{vmatrix}=(1-t)e^t
\end{equation}
\begin{equation}
W_1[y_1,y_2]=\begin{vmatrix}
{0}&{y_{2}(t)}\\
{1}&{y_{2}'(t)}\\
\end{vmatrix}=-t
\end{equation}
\begin{equation}
W_2[y_1,y_2]=\begin{vmatrix}
{y_{1}(t)}&{0}\\
{y_{1}'(t)}&{1}\\
\end{vmatrix}=e^t
\end{equation}
Then:
\begin{equation}
Y(t)=y_{1}\int{\frac{W_1[y_1,y_2]*g(t)}{W[y_1,y_2]}dt}+y_{2}\int{\frac{W_2[y_1,y_2]*g(t)}{W[y_1,y_2]}dt}
\end{equation}
\begin{equation}
Y(t)=e^t\int{\frac{-t*2(1-t)e^{-t}}{(1-t)e^t}dt}+t\int{\frac{e^t*2(1-t)e^{-t}}{(1-t)e^t}dt}
\end{equation}
\begin{equation}
Y(t)=-2e^t\int{\frac{t}{e^{2t}}dt}+2t\int{\frac{1}{e^t}dt}
\end{equation}
\begin{equation}
Y(t)=-2e^t((-\frac{t}{2}-\frac{1}{4})e^{-2t})+2t(-2e^{-t})
\end{equation}
\begin{equation}
Y(t)=(\frac{1}{2}-t)e^{-t}
\end{equation}
Therefore the particular solution of the given nonhomogeneous equation is
$(\frac{1}{2}-t)e^{-t}$
(By setting constants to 0)
