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## Homework Statement

Solve, [tex]u_{t} = u_{xx}c^{2}[/tex]

given the following boundary and initial conditions

[tex]u_{x}(0,t) = 0, u(L,t) = 0[/tex]

[tex]u(x,0) = f(x) , u_{t}(x,0) = g(x)[/tex]

## Homework Equations

[tex]u(x,t) = F(x)G(t)[/tex]

## The Attempt at a Solution

I solved it, I am just not sure if it is right.

[tex]u(x,t) = \sum_{n=1}^\infty(a_{n}cos(\lambda_{n}t) + b_{n}sin(\lambda_{n}t))cos((n-\frac{1}{2})\frac{\pi}{L}x)

, \lambda_{n} = (n-\frac{1}{2})\frac{\pi}{L}c [/tex]

[tex]a_{n} = \frac{2}{L}\int_0^L f(x)cos((n-\frac{1}{2})\frac{\pi}{L}x)dx,

b_{n} = \frac{4}{(2n-1)c\pi}\int_0^L g(x)cos((n-\frac{1}{2})\frac{\pi}{L}x)dx

[/tex]

Can someone please verify this for me?

Thanks in advance,

KEØM

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