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 $\frac{\pi}{2}$

 

[Mark44]

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

 

[Mark44]

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?

 

[Mark44]

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?