Partial differential equation

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Please see the attached file.

PDE Utt = Uxx+sin(3x) 0<x<1 0<t<

BC's U(0,t)=0
U(1,t)= 0

IC's U(x,0)= 2sin(2x)
Ut(x,0)= 0

We first must consider the homogenous equation:

We start as always with separating the wave equation:

Plugging it back into the homogenous equation we obtain:

Each side is independent of the other (since each side is a function of a single independent variable).

This can occur if an only if both sides equal the same constant number.

Hence:

The spatial equation becomes:

Case 1:

The general solution of the equation is:

Using boundary conditions we have:

Hence for we get only the trivial solution.

Case 2:

The general solution of the equation is:

Using boundary conditions we have:

Hence for we get only the trivial solution.

So we are left with case 3:

The general solution of the equation is:

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