Vibrating string displacement, partial differential problem

[leoflc]

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!

 

[mighty2000]

Thank you for your question. Your approach seems to be on the right track. However, there are a few things to consider in order to solve for u(x,t) with the given conditions.

Firstly, you are correct in using the separation of variables method to solve the partial differential equation. By doing this, you have obtained two ordinary differential equations for F(x) and G(t). However, in order to solve for u(x,t), you need to combine these two equations and apply the boundary conditions and initial conditions given. This will help you to determine the constants in your equations and ultimately solve for u(x,t).

Secondly, you have correctly obtained the solution for F(x) as F(x) = C*sin(Px). However, for G(t), you should get G(t) = Ae^(-beta/2)t * cos(Pt) + Be^(-beta/2)t * sin(Pt), where A and B are constants to be determined using the initial conditions.

Lastly, in order to apply the initial conditions, you need to substitute the given values of u(x,0) = f(x) and u_t(x,t) = 0 into the solution for u(x,t) obtained from combining the two equations. This will give you two equations with two unknown constants (A and B) that you can solve for.

I hope this helps you to solve for u(x,t). Good luck with your work!