# Partial Differential Equation Boundary Conditions

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Find the general solution of the wave equation U(tt) = U(xx) subject to the boundary conditions u(0,t) = u(1,t) = 0.

Then find the unique solution of the wave equation subject to the initial conditions:

u(x,0) = 2sin3pix and ut(x,0) = 5 sinpix

Thanks

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The equation is:

(1.1)

The boundary conditions are:

(1.2)

And initial conditions:

(1.3)

And:

(1.4)

We start with separating the equation by defining:

(1.5)

The functions are completely independent of each other and therefore:

(1.6)

Plugging this back into (1.1) and dividing by we obtain:

(1.7)

Each side of (1.7) is completely independent of the other side. The left hand side is a function of t while the right hand side is a function of x. Since (1.7) must ...

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