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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

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The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.
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 ...

Solution Summary

The expert examines partial differential equation boundary conditions.

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