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## Homework Statement

Solve

[tex]u_{tt}(x,t)+2u_{t}(x,t)-u_{xx}(x,t)=18\sin\left(\dfrac{3\pi x}{l}\right)[/tex]

[tex]\omega=\frac{\pi n c}{l}[/tex]

Boundary conditions

[tex]u(0,t)=u(l,t)=0[/tex]

[tex]l<\pi[/tex]

Initial conditions

[tex]u(x,0)=u_t(x,0)=0[/tex]

## Homework Equations

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The general inhomogeneous damped wave equation is

[tex]u_{tt}(x,t)+2\mu u_{t}(x,t)-c^{2}u_{xx}(x,t)=p(x,t)[/tex]

## The Attempt at a Solution

By separation of variables the position dependent part is

[tex]f(x)=a\cos(kx)+b\sin(kx)[/tex]

and the time dependent part auxilliary equation is

[tex]\lambda^2+2\mu\lambda+\omega^2=0[/tex]

which takes one of three expressions depending on the sign of [itex]\mu-\omega[/itex]

So from what is given I would say [itex]\mu=1[/itex] and [itex]c^2=1[/itex] and with

[itex]\omega=\frac{\pi n c}{l}[/itex] and [itex]l<\pi[/itex] then [itex]\mu<\omega[/itex] which is weak damping

[tex]g_{n}(t)=\left(A_{n}\cos\left(\Omega t\right)+B_{n}\sin\left(\Omega t\right)\right)e^{-\mu t}[/tex]

where [tex]\Omega=\sqrt{\omega^{2}-\mu^{2}}[/tex]

Is this correct so far?