I've been trying to solve the following ODE
$$ 4 x\left(y(x) \times \frac{\partial^{2} y(x)}{\partial x^{2}}+\left(\frac{\partial y(x)}{\partial x}\right)^{2}\right)+2 y(x) \times \frac{\partial y(x)}{\partial x}+y(x)^{2} x k \times \frac{\partial y(x)}{\partial x}=0, $$
with $x = \psi^2/t, \ u(\psi,t)=y(x), \ u(\psi,0)=N_0 \delta(\psi)$, but unfortunately I'm at a loss. Any ideas?
1. y = y(x) or y = y(x, t)?
The differential equation establishes that y is a function only of x, ie: y = y(x).
However, your boundary condition establishes that y is a function of something else, maybe time? y = y(x, t)?
2. What kind of answer?
Are you looking for analytical or numerical solutions?
3. Possible solution path
This kind of problem is usually solved by integral transform techniques or by finite elements. However, you would have to explain better the problem and what you are trying to achieve in order to decide which one would be the best.