Applying Initial Conditions

MathematicalPhysics

I need to find all the separated solns of

$$x^2 \frac{\partial^2 u}{\partial x^2} + x\frac{\partial u}{\partial x} + \frac{\partial^2 u}{\partial y^2} = 0$$

in the strip $${(x,y) : 0 < y < a, -\infty < x < \infty }$$
the separated solns must also satisfy u = 0 on both the edges, that is, on y=0 and y=a for all values of x.

Iv got the general solutions to be..

$$X(x) = Dlnx + C , (k = 0)$$
$$X(x) = Dx^{n} + Cx^{-n} , (k \neq 0)$$

and

$$Y(y) = A\cos{ky} + B\sin{ky} , (k \neq 0)$$
$$Y(y) = Ay + B , (k = 0)$$

where k is just the constant iv let the two bits equal when I separated the variables. (well -k^2 actually).

I just need help interpreting the conditions to sort out the constants..I think!

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Galileo

Homework Helper
I haven't checked your answer, but if it is correct then, since $u(x,y)=X(x)Y(y)$, the boundary conditions say:

$$u(x,0)=X(x)Y(0)=0$$
and
$$u(x,a)=X(x)Y(a)=0$$

So $Y(0)=Y(a)=0$

For example: if k=0, then applying the boundary condition at y=0 gives:
$$Y(0)=B=0$$

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