# PDE Separation of variables

## Homework Statement

Solve the problem.
utt = uxx 0 < x < 1, t > 0
u(x,0) = x, ut(x,0) = x(1-x), u(0,t) = 0, u(1,t) = 1

## 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)
ut(x,t) = X(x)T'(t) ux(x,t) = X'(x)T(t)
utt(x,t) = X(x)T"(t) uxx(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 = ω2
X(x) = Acosh ωx + Bsinh ωx
X(0) = Acosh ω(0) + Bsinh ω(0) = 0
= Bsinh ω(0) = 0 ------> B = 0

λ < 0 ---------> λ = -ω2
X"(x) = λX(x) --------> X"(x) = -ω2X(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) = ω2T(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!

HallsofIvy
Homework Helper

## Homework Statement

Solve the problem.
utt = uxx 0 < x < 1, t > 0
u(x,0) = x, ut(x,0) = x(1-x), u(0,t) = 0, u(1,t) = 1

## 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)
ut(x,t) = X(x)T'(t) ux(x,t) = X'(x)T(t)
utt(x,t) = X(x)T"(t) uxx(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 = ω2
X(x) = Acosh ωx + Bsinh ωx
X(0) = Acosh ω(0) + Bsinh ω(0) = 0
= Bsinh ω(0) = 0 ------> B = 0

λ < 0 ---------> λ = -ω2
X"(x) = λX(x) --------> X"(x) = -ω2X(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) = ω2T(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!

## The Attempt at a Solution

Other than the fact that you mean "sin ωx", not "sin ux", that's good. Now you have to try to make that fit the "initial vaue conditions", u(x,0)= 0 and $u_t(x, 0)= x(1- x)$.

You won't be able to do that with just a single "ω" so since this is a linear equation try, instead, a sum:
$$u(x,t)= \sum_{m=0}^\infty (cos(\pi/2 + 2m\pi)t + Dsin(\pi/2 + 2m\pi)t)sin(\pi/2 + 2\pi)x$$

Sorry to be so dense, but I get lost at this point.

I think I am then suppose to do

ut=X(x)[-C(∏/2 + 2m∏)sin(∏/2 + 2m∏)t + D(∏/2 + 2m∏)cos(∏/2 + 2m∏)t)
ut(x,0) = D(∏/2 + 2m∏)cos(∏/2 + 2m∏) = x(1-x) ----> D ≠ 0

t = 0 f(x) = ∑ Dsin (∏/2 + 2m∏)t

u(x,t) = ∑ D sin ((∏/2 + 2m∏)t sin (∏/2 + 2m∏)x

Am I on the doing this correctly? Do I then do the integral from 0 to m∏?

Thanks!