Solving Inhomogeneous Wave PDE with Separation of Variables on (0,pi)

[jacoleen]

Problem:

Use separation of variables to solve
u_tt = u_xx-u;
u(x, 0) = 0;
u_t(x, 0) = 1 + cos³ x;
on the interval (0, pi), with the homogeneous Dirichlet boundary conditions.

Question:

I know how to use separation of variables, but can`t figure out what to do with the u in the equation u_tt=u_xx=u..any hints please?

 

[MathematicalPhysicist]

Duhammel's principle:
http://en.wikipedia.org/wiki/Duhamel's_principle

seperate this problem to the next problem:

v_tt-v_xx=0
v(x,0)=0
v_t(x,0)=1+cos^3(x)

and
w_tt-w_xx= -u(x,t)
w(x,0)=0
w_t(x,0)=0

So u(x,t)=w(x,t)+v(x,t)

For the w use Duhamel, for v separation of variables.

 

[MathematicalPhysicist]

Never mind what I wrote, you can solve it by seperation:

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

u_tt = T''(t) X(x) = T(t)X''(x)-TX
divide by XT and get T''/T= X''/X-1
one side depends on t the other side on x so both of them are constant.

 

[jacoleen]

oh wow, I can`t believe i didnt think of dong that..thanks! :D

 

[jacoleen]

one more question, when i separate my variables i get
X(x) = Ae^{(1-lamba2)1/2x} + Be^{-(1-lamba2)1/2x}
T(t) = Csin(lamba*t)+Dcos(lamba*t),
but when I solve it I get all my coefficients equal to zero..are these the right equations?