# PDE: Wave equation with first order derivative

1. Aug 6, 2013

### talolard

1. The problem statement, all variables and given/known data

Solve using separation of variables utt = uxx+aux
u(0,t)=u(1,t)=0
u(x,0)=f(x)
ut=g(x)

3. The attempt at a solution
if not for the ux 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!!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Aug 6, 2013

### 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 $X(x)$.

Last edited: Aug 6, 2013
3. Aug 6, 2013

### haruspex

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