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## Homework Statement

A damped vibrating string of length 1, that satisfies

u_tt = u_xx - ([tex]\beta[/tex])u_t

with the boundary conditions:

u(0,t)=0

u(1,t)=0

initial conditions:

u(x,0)=f(x)

u_t(x,t)=0

solve for u(x,t) if [tex]\beta[/tex]^2 < 4Pi^2

## The Attempt at a Solution

if u(x,t)=F(x)G(t)

So by using partial differential equations, I got:

G''+[tex]\beta[/tex]G'+GP^2=0

and

F''+FP^2=0

I solved for F(x) with B.Cs and got:

F(x)=C*Sin(P*x), where C is a const.

when I tried to solve for G(t), I got a long equation with 2 constants in there.

If I try to solve for u(x,t) by using the F(x)G(t), I will have something with three unknow constants.

Am I on the right track?

I'm not sure how/when to apply I.Cs and [tex]\beta[/tex] to solve for u(x,t).

Thank you very much!