# Solving Partial differential equation

AHSAN MUJTABA
Homework Statement:
solve the equation for $$\frac{\partial^{2}\psi(\sqrt{g} y)}{\partial y^{2}}=(\frac{sin^{2}(\sqrt{g}y)}{g}-\frac{2E}{\omega})\psi(\sqrt{g}y)$$.
Relevant Equations:
The boundary conditions are $$\psi(\frac{\pi}{2})=0$$ and $$\psi(\frac{-\pi}{2})=0$$. I know the values of ##\frac{E}{\omega}##
I have tried to do it in standard way by integrating in PDE's but it turned out that ##\psi## is a function of y, so now I have no clue to start this. I know the range of ##\sqrt {g}y## from ##\frac{-\pi}{2}## to ##\frac{\pi}{2}##

Mentor
Homework Statement:: solve the equation for $$\frac{\partial^{2}\psi(\sqrt{g} y)}{\partial y^{2}}=(\frac{sin^{2}(\sqrt{g}y)}{g}-\frac{2E}{\omega})\psi\sqrt{g}y$$.
Relevant Equations:: The boundary conditions are $$\psi(\frac{\pi}{2}=0$$ and $$\psi(\frac{-\pi}{2})=0$$. I know the values of ##\frac{E}{\omega}##

I have tried to do it in standard way by integrating in PDE's but it turned out that ##\psi## is a function of y, so now I have no clue to start this. I know the range of ##\sqrt {g}y## from ##\frac{-\pi}{2}## to ##\frac{\pi}{2}##
Since ##\phi## is apparently a function of y alone, it seems to me that you're not dealing with a partial differential equation -- just an ordinary differential equation. An integrating factor might work.

AHSAN MUJTABA
Since ##\phi## is apparently a function of y alone, it seems to me that you're not dealing with a partial differential equation -- just an ordinary differential equation. An integrating factor might work.
If I just double integrate this equation on y, you mean that?

AHSAN MUJTABA
I was thinking of some series solutions but I have to solve it properly

Mentor
If I just double integrate this equation on y, you mean that?
No, that's not what I mean. I would first do a substitution -- Let ##u = \sqrt g y## -- and then get a DE that looks like this:
##\frac{d^2 \phi(u)}{du^2} - (\frac{\sin^2(u)} u - \frac {2E} \omega)\phi(u) = 0##
It won't look exactly like this, because there are some factors that I've omitted that come from the chain rule.

This is an ordinary differential equation, not a PDE, but it's nonlinear. I don't have any ideas for attacking it at the moment.

AHSAN MUJTABA
I have tried to attack it with the same strategy but since ##\phi(u)## at the end creates a high problem for solving this.

AHSAN MUJTABA
Can I substitute appropriately to make it a first order ode?

Mentor
Can I substitute appropriately to make it a first order ode?
No, because you have ##\phi''## and ##\phi## in the equation.

AHSAN MUJTABA
Is there going to be a power series solution?