- #1

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[tex] \frac{\partial u}{\partial t} - k \frac{\partial^2 u}{\partial x^2} = 0 [/tex] for 0 <x < pi, t > 0

[itex] u(0,t) = u(\pi,t) = 0 [/itex]

[itex] u(x,0)= \Sin^2 x [/itex]

let u (x,t) = X(x) T(t)

[tex] \frac{X''}{X} = \frac{T'}{T} = -\lambda = \mu^2 [/tex]

also lambda must be positive (imaginary solution)

[tex] X(x) = C_{1} \cos(\mu x} + C_{2} \sin(\mu x) [/tex]

using the boundary conditons

C1 = 0 and let u = n some positive integer

[tex] X_{n} (x) = \sin(\mu x) [/tex]

also solution for T is

[tex] T_{n} (t) = e^{-k \mu^2 t} [/tex]

now for T(0) = sin^3 x

sin ^3 x = 1 ?

SO x must be pi/2? since 0 <x < pi

i dont understand how to proceed from here

I know that i have to use some infinite series hereafter...

Please help on this!

Thank you for your help!