Use D'Alembert's solution to solve the following

subject to the initial conditions and .

Note that with and

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- Aug 16th 2010, 05:51 AMJoernEPDE using D'Alembert's Solution
Use D'Alembert's solution to solve the following

subject to the initial conditions and .

Note that with and - Aug 16th 2010, 07:36 AMJester
With your change of variables

gives

Integrating twice gives

or

Imposing the two boundary conditions gives

.

Differentiating the first and then solving for F' and G' gives

.

Thus, .

Finally, we have

but imposing the first BC (1) gives , so the final solution is

- Aug 24th 2010, 05:20 AMJoernE
- Aug 24th 2010, 06:34 AMJester
I used it after making the change of variables. Let me derive my solution with the D'Alembert solution.

First the D'Alembert solution applies for

But your PDE has a nonhomogeneous term.

So we'll find a particlar solution - works. So let where now satisfies

,

D'Almbert solution

and your IC's gives and

so

. - Aug 25th 2010, 05:00 AMJoernEQuote:

so we'll find a particular solution works

- Aug 25th 2010, 05:35 AMJester