How Do You Solve the Vibrating String Problem Using Differential Equations?

[hydralisks]

Homework Statement

Find a formal solution to the vibrating string problem..

alpha=4, 0<x<pi t>0
u(0,t)=u(pi,t)=0 t>0

f(x)= x^2(pi-x)
g(x)=0

Homework Equations

u(x,t) = sum[a cos(alpha*n*t/L + b sin(alpha*n*t/L)*sin(n pi x / L)

Fourier series for sine

The Attempt at a Solution

a = 2/pi * integral(x^2 (pi-x) * sin(nx) dx) from 0 to pi
= [-2((n pi)^2 - 2)(-1)^n - 4 + 2((n pi)^2 - 6)(-1)^n] / n^3 ... by breaking it into addition of 2 integrals and using the tabular method of integration

b = 2/pi * integral(0 * sin(nx) dx) from 0 to pi
= 0however, the answer is
u(x,t) = sum[ 4/n^3 * [2(-1)^(n+1) - 1] * cos2nt sinnxDid I do something wrong or is the answer in a different form? I don't see how to get from my answer to the correct one.

 

[mighty2000]

Your attempt at a solution looks correct. The only difference between your answer and the given solution is the use of a different trigonometric identity. The given solution uses the identity sin(2x) = 2sin(x)cos(x), while you used sin(nx) = n*sin(x).

If you substitute the given solution into the Fourier series formula for sine, you should get the same answer as your attempt. However, both forms are correct and can be used interchangeably.