Problem source

\begin{questionparts}
\item Find functions ${\rm a}(x)$ and ${\rm b}(x)$ such that $u=x$ and
  $u=\e^{-x}$
both satisfy the equation 
$$\dfrac{\d^2u}{\d x^2} +{\rm a}(x) \dfrac{\d u}{\d x} + {\rm b} (x)u=0\,.$$ 
For these functions ${\rm a}(x)$ and ${\rm b}(x)$, write 
down the  general solution of the equation.
Show that  the substitution $y= \dfrac 1 {3u} \dfrac {\d u}{\d x}$
transforms the equation
\[
\frac{\d y}{\d x} +3y^2 + \frac {x} {1+x} y = \frac 1 {3(1+x)}
\tag{$*$}
\]
into
\[
\frac{\d^2 u}{\d x^2} +\frac x{1+x} \frac{\d u}{\d x} - \frac 1 {1+x}
u=0
\]
and hence show that the solution of equation ($*$) that satisfies
$y=0$ at $x=0$ is given by
 $y = \dfrac{1-\e^{-x}}{3(x+\e^{-x})}$.
\item
Find the solution of the equation
$$
\frac{\d y}{\d x} +y^2 + \frac x {1-x} y = \frac 1 {1-x}
$$
that satisfies $y=2$ at $x=0$.

\end{questionparts}