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KEØM
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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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