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7.28, §4 Solve

Expand all variables in the problem for u in a Fourier series:

Put into P.D.E. u_tt = a² u_xx:

Because of the choice of the X_n:

Put into I.C.s u(x,0) = f(x) and u_t(x,0) = g(x):

Solve this O.D.E. and initial condition for C_n:

Characteristic polynomial:

Note: I would have gotten this same solution if I would have solved for C(t) in the previous section.

Initial conditions:

Next: 7.28, §5 Total Up: 7.28 Previous: 7.28, §3 Eigenfunctions