Verify that the given functions y1 and y2 satisfy the corresponding homogeneous equation then find a particular solution of the given nonhomogeneous equation
$(1-t) y^{\prime \prime}+t y^{\prime}-y=2(t-1)^{2} e^{-t}, 0<t<1$
$y_{1}(t)=e^{t}, y_{2}(t)=t$
$(1-t) y^{\prime \prime}+t y-y=2(t-1)^{2} e^{-t}, 0<t<1 ; y_{1}(t)=e^{t}, y_{2}(t)=t$
$\left\{\begin{array}{l}{y_{1}(t)=e^{t}} \\ {y_{1}^{\prime}(t)=e^{t}} \\ {y_{1}^{\prime \prime}(t)=e^{t}}\end{array}\right.$ and $\left\{\begin{array}{l}{y_{2}(t)=t} \\ {y_{2}^{\prime}(t)=1} \\ {y_{2}^{\prime \prime}(t)=0}\end{array}\right.$
$\qquad$Substitute back into the homogeneous equation
$(1-t) y^{\prime \prime}+t y^{\prime}-y=0$
$y^{\prime \prime}+\frac{t}{1-t}-\frac{1}{1-t}=-2(t-1) e^{-t}$
$\qquad$Then
$p(t)=\frac{t}{1-t}, q(t)=-\frac{1}{1-t}, g(t)=-2(t-1) e^{-t}$
$W\left[y_{1}, y_{2}\right](t)=\left|\begin{array}{ll}{y_{1}(t)} & {y_{2}(t)} \\ {y_{1}^{\prime}(t)} & {y_{2}^{\prime}(t)}\end{array}\right|=(1-t) e^{t}$
Since the particular solution has the form:
$Y(t)=u_{1}(t) y_{1}(t)+u_{2}(t) y_{2}(t)$
and
\begin{equation}
\label{eq:D1}
u_{1}(t)=-\int \frac{y_{2}(t) g(t)}{W\left[y_{1}, y_{2}\right](t)} d t
\end{equation}
\begin{equation}
\label{eq:D2}
=-\int \frac{t \cdot\left(-2(t-1) e^{-t}\right)}{(1-t) e^{t}} d t
\end{equation}
\begin{equation}
\label{eq:D3}
=-2 \int t e^{-2 t} d t
\end{equation}
\begin{equation}
\label{eq:D4}
=\left(t+\frac{1}{2}\right) e^{-2 t}
\end{equation}
\begin{equation}
\label{eq:D5}
u_{2}(t)=\int \frac{y_{1}(t) g(t)}{W\left[y_{1}, y_{2}\right](t)} d t
\end{equation}
\begin{equation}
\label{eq:D6}
=\int \frac{e^{t} \cdot\left(-2(t-1) e^{-t}\right)}{(1-t) e^{t}} d t
\end{equation}
\begin{equation}
\label{eq:D7}
=2 \int e^{-t}
\end{equation}
\begin{equation}
\label{eq:D8}
=-2 e^{-t}
\end{equation}
Therefore,
$Y(t)=\left(t+\frac{1}{2}\right) e^{-2 t} \cdot e^{t}+\left(-2 e^{-t}\right) \cdot t=\left(\frac{1}{2}-t\right) e^{-t}$
The general solution is:
\begin{equation}
\label{eq:D9}
y(t)=y_{c}(t)+Y(t)
\end{equation}
\begin{equation}
\label{eq:D10}
=c_{1} e^{t}+c_{2} t+\left(\frac{1}{2}-t\right) e^{-t}
\end{equation}
Therefore,the particular solution of the given nonhomogeneous equation is
$Y(t)=\left(\frac{1}{2}-t\right) e^{-t}$
