PDE: Wave equation with first order derivative

[talolard]

Homework Statement

Solve using separation of variables u_tt = u_xx+au_x
u(0,t)=u(1,t)=0
u(x,0)=f(x)
u_t=g(x)

The Attempt at a Solution

if not for the u_x I'd set
U=XT
such that X''T=TX'' and using initial conditions get a solution.

In my case I get T''X=T(aX'+X'') which is still solvable but highly unpleasant. The question comes from an old exam and I wonder if their is something I'm not seeing?

Thanks!

 

[pasmith]

Consider the substitution u(x,t) = e^{\beta x}v(x,t), for a suitable value of \beta.

EDIT: Forget that, it doesn't simplify as I thought it would. But if you separate the variables as is, you end up with a perfectly normal 2nd order linear ODE with constant coefficients for X(x).

 

[haruspex]

In my case I get T''X=T(aX'+X'') which is still solvable but highly unpleasant.

Is it stated anywhere that the solution will be pleasant?
Post your subsequent working down to the point where you find it suspiciously messy.