solving a second order nonlinear pde I would like to solve the following PDE,
$$f_{y}^{2} = 2 f f_{yy}$$
where $f= f(x,y)$ is a real function of two variables $x,y$.
My solution
: derivative of $f_{y}^{2}$ with respect to $y$ is itself, so $f_{y}^{2}= c(x)e^{y}$ and 
$$f_{y} = c(x) e^{\frac{y}{2}}$$
so, 
$f(x,y) = 2 c(x) e^{\frac{y}{2}} + d(x)$
but this does not satisfy in the equation $f_{y}^{2} = 2 f f_{yy}$ .
thanks for your help.

$$
f_yf \left(\frac{f_y}{f} -2\frac{f_{yy}}{f_y}\right) = 0
$$
Thus we can obtain
$$
\dfrac{\partial}{\partial y}\ln f -2\dfrac{\partial }{\partial y}\ln f_y = 0
$$
Or
$$
\ln \left(\frac{f}{f_y^2}\right) = g(x)
$$
So you get
$$
f = g_2(x) f_y^2\implies f_y = \sqrt{\frac{f}{g_2(x)}}
$$
