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## Homework Statement

Solve the problem.

u

_{tt}= u

_{xx}0 < x < 1, t > 0

u(x,0) = x, u

_{t}(x,0) = x(1-x), u(0,t) = 0, u(1,t) = 1

## Homework Equations

## The Attempt at a Solution

Here is what I have so far but I'm not sure if I am on the right path or not.

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

u

_{t}(x,t) = X(x)T'(t) u

_{x}(x,t) = X'(x)T(t)

u

_{tt}(x,t) = X(x)T"(t) u

_{xx}(x,t) = X"(x)T(t)

X(x)T"(t) = X"(x)T(t)

T"(t)/T(T) = X"(x)/X(x) = λ

T"(t) = λT(t) X"(x) = λX(x)

λ = 0 -----> X(x) = Ax + B

b.c. u(0,t) = A(0) + B = 0 --------> B = 0

u(1,t) = A(1) + B = 1 --------> A = 1

λ > 0 --------> λ = ω

^{2}

X(x) = Acosh ωx + Bsinh ωx

X(0) = Acosh ω(0) + Bsinh ω(0) = 0

= Bsinh ω(0) = 0 ------> B = 0

λ < 0 ---------> λ = -ω

^{2}

X"(x) = λX(x) --------> X"(x) = -ω

^{2}X(x)

X(x) = Acosωx + Bsinωx

X(0) = Acosω(0) + Bsinω(0) = 0 --------> A = 0

X(1) = Acosω(1) + Bsinω(1) = 1

X(1) = Bsinω = 1 B ≠ 0

ω = ∏/2 + 2m∏ for any interger m

T"(t) = ω

^{2}T(t)

T"(t) = C cosωt + Dsinωt

u = (C cosωt + Dsinωt)sinux

Okay this is all I have. Am I on the right path and where do I go from here?

Thanks!