Find the general solution to

The solution is

How is this obtained?

Thanks.

I solved the first part:

Should that be in the integral the partial derivative or just derivative?

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- December 23rd 2010, 02:22 PMdwsmithu_{xx}-u=0
Find the general solution to

The solution is

How is this obtained?

Thanks.

I solved the first part:

Should that be in the integral the partial derivative or just derivative?

- December 23rd 2010, 03:29 PMpickslides
I think of this problem as find the solution such that

As you only have a partial with respect to the same variable then the solution is similar to that of a 2nd order ODE. I.e has the solution and

The difference in your problem is that and need to be introduced but are constant with respect to - December 23rd 2010, 03:31 PMpickslides
- December 23rd 2010, 03:31 PMdwsmith
- December 23rd 2010, 03:32 PMdwsmith
- December 23rd 2010, 03:36 PMdwsmith
I understand your method which is simple but I would like to finish the problem how I started it. If you wouldn't mind, how should I finish it off?

Thanks.

If I just integrate, I wouldn't obtain the e^{-x} so I am not sure how to proceed. - December 23rd 2010, 03:46 PMpickslides
- December 23rd 2010, 03:49 PMdwsmith
I just looked it up. Integrating factors aren't restricted to first order linear equations.

- December 23rd 2010, 04:21 PMpickslides
I would say

Gives characteristic equation in the form and therefore seek a solution in the form of

As we have a PDE and can also be separate functions of . - December 23rd 2010, 04:45 PMdwsmith
Now, I am looking to find the sol. of which satisfies the auxiliary conditions

How do I incorporate this into the solution?

- December 23rd 2010, 09:00 PMdwsmith
Is this the correct procedure?

By adding the two equations and simplifying, we obtain

Now, solving for I get