Derive the weak form for nonlinear problem. Let the equation be $$\frac{d^2 y}{dx^2}=\frac{y}{1+y}$$
For finite element formulation how to get the weak form? The major problem being the nonlinear rhs.`
 A: Let me rewrite your expression as
\begin{align}
y_{xx} = \frac{y}{1+y}\Leftrightarrow y_{xx} +yy_{xx} -y = 0
\end{align}
Since
\begin{align}
\Bigl(\frac{y^2}{2}\Bigr)_{xx} =  (yy_x)_x = y_x^2+yy_{xx} \Leftrightarrow yy_{xx} = 
\Bigl(\frac{y^2}{2}\Bigr)_{xx}-y_x^2
\end{align}
you can rewrite the first expression as
\begin{align}
 y_{xx} +yy_{xx} -y = 0
\Leftrightarrow
y_{xx} +\Bigl(\frac{y^2}{2}\Bigr)_{xx}-y_x^2-y = 0
\end{align}
Assume, that $\phi^i$ are our (standard) testfunctions (which vanish on $\partial \Omega$). For the weak formulation we project onto the testspace. Let $\Omega$ be our domain, we then have for all $i$
\begin{align}
&\int_{\Omega}\Bigl[ y_{xx} +\Bigl(\frac{y^2}{2}\Bigr)_{xx}-y_x^2-y \Bigr]\phi^i\,dx = 0\\
\Leftrightarrow&
-\int_{\Omega}\Bigl[y_x+\frac{y^2}{2}\Bigr]\phi^i_x\, dx -\int_\Omega \Bigl[y^2_x+y \Bigr] \phi^i\,dx=0
\\
\Leftrightarrow &
\int_{\Omega}\Bigl[y_x+\frac{y^2}{2}\Bigr]\phi^i_x\, dx +\int_\Omega \Bigl[y^2_x+y \Bigr] \phi^i\,dx=0
\end{align}
Now you have only derivatives of order zero and one and can use $y\approx y^N$ as it is always done with FEM.
