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**linalg123** **Question:** Find the temperature u(x,t) in a rod of length L if the initial temp is f(x) throughout and the ends are x=0 and x=L are insulated.

Solve if L=2 and f(x) = {x , 0 < x <1

{0 , 1 < x <2

**attempt: **

k d^2(u)/dx^2 = du/dt ; 0<x<L, t>0

Insulated ends means: u(0,t)=0 u(L,t) = 0

u(x,0)= f(x) ; 0 <x<L

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

d^2(u)/d(x)^2 = TX'' ; du/dt = XT'

kTX'' = XT'

X''/X = T'/(kT) = -λ^2

X'' + λ^2X = 0

T' + λ^2kt = 0

X'(0)=0 X'(L)=0

for λ=0 , X(x) = ax+b

but from the boundary condition a=0 , X(x)=b

for λ^2 > 0

X= acosλx + bsinλx

X' = λ(-asinλx +bcosλx)

this is where i am getting stuck. any help would be great, thanks