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For the 1-D wave equation, the d’Alembert solution is u(t, x) = f (x + ct) + g(x − ct) where f , g are each a function of 1 variable.

Suppose c = 1 and we know f (x) = x^2 and g(x) = cos 2x for x > 0.

Find u(t, x) for al l t, x ≥ 0 if you are also given the BC: u ≡ 1 at x = 0, all t.

2. See problem statement

3.

Do we simply "plug" into the D'alembert solution?

I started with:

u(t, x) = f (x + ct) + g(x − ct)

so: f(x)=x^2 or f(x)=(x+t)^2

so: g(x)=cos2x or g(x)=cos(2x-2t)

next: plug f(x), g(x) into u(t,x)

u(t,x)=x^{2}+2xt+t^{2}+cos(2x-2t)

Is this all that needs to be done? I am not sure where to go from here or what to do with the BC of u(t,0)=1. Please help.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# D'Alembert Problem for 1-D wave equation

Can you offer guidance or do you also need help?

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