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## Homework Statement

Find the solution u, via the Fourier sine/cosine transform, given:

[tex]u_{tt}-c^{2}u_{xx}=0[/tex]

IC: [tex]u(x,0) = u_{t}(x,0)=0[/tex]

BC: u(x,t) bounded as [tex]x\rightarrow \infty , u_{x}(0,t) = g(t)[/tex]

**2. The attempt at a solution**

Taking the Fourier transform of the PDE, IC and BC:

[tex]U_{tt}-c^{2}(i\lambda)^{2}U=0[/tex]

[tex]U_{tt}+c^{2}\lambda^{2}U=0[/tex]

which is an ODE in t, so two linearly independent solutions of the homogeneous equation are [tex]sin(\lambda ct)[/tex] and [tex]cos(\lambda ct)[/tex].

If I take a linear combination of these two solutions, I get zero constants, which eventually leaves me with u = 0 as a solution, or at least one that's only x-dependent.

But if u is x-dependent, [tex]U_{tt} = 0 \Rightarrow c^{2}\lambda^{2}U=0 \Rightarrow U = 0[/tex]

which still leaves me with u = 0 as a solution, and, no offense, for a homework problem, that's kind of lame. That's why I'm suspicious, and asking to see if I'm doing it right.

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