Inhomogeneous Wave Equation: How to Solve using Separation of Variables?

[ARTjoMS]

u''tt=a^2*u''xx + t*x 0<x<l; t>0

u(0,t)=u(l,t)=0
u(x,0)=u't(x,0)=0

http://eqworld.ipmnet.ru/en/solutions/lpde/lpde202.pdf

^^Here i found how to solve this problem using Green's function, however i am told to solve this using the method of separation of variables. But i cannot find any theory or examples for inhomogeneous wave equation.

Please help me.
Thank you!

 

[jackmell]

Well normally when you do separation of variables you let:

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

How about trying other expressions with the variables separated. For example, try:

u(x,t)=xX(x)tT(t)

or:

u(x,t)=\frac{XT}{xt}

or:

u(x,t)=\frac{xt}{XT}

 

[jasonRF]

In general separation of variables applies to homogeneous problems, and you have a "forcing term" (t*x in your example). Sometimes you can be clever and muck with your original problem to place it in a form suitable for separations of variables directly, but I don't know of any general procedure for doing this.

The usual procedure, at least for "nice" equations like the wave equation, is to use separation of variables on the homogeneous problem in order to find the natural eigenfunctions. One then expands both the solution and the forcing term in those eigenfunctions, and solve for the resulting coefficients.

Another approach for this problem is to use the method of characteristics, leading to the well-known solution known as the d'Alembert solution.

good luck,

jason