D'alembert solution for the semi-infinite string

[bobred]

Homework Statement

Find the solution of the wave equation using d'Alembert solution.

Homework Equations

[itex]u(0,t)=0[/itex][/B] and [itex]u(x,0)=0[/itex]
[tex]u_t(x,0)=\frac{x^2}{1+x^3}, \, x\geq0[/tex]
[tex]u_t(x,0)=0, \, x<0[/tex]

The Attempt at a Solution

For a semi infinite string we have the solution
[tex]u(x,t)=\frac{1}{2}\left( a(x-ct)+a(x+ct)-a(-x-ct)-a(-x+ct) \right)+\frac{1}{2c}\left( \int^{x+ct}_{x-ct} dy\, b(y) - \int^{-x+ct}_{-x-ct} dy\, b(y) \right)[/tex]
with [itex]u(x,0)=a(x)=0[/itex] so
[tex]u(x,t)=\frac{1}{2c}\left( \int^{x+ct}_{x-ct} dy\, b(y) - \int^{-x+ct}_{-x-ct} dy\, b(y) \right)[/tex]
where

[tex]b(y)=\frac{y^2}{1+y^3}[/tex]

Is this right?

 

[bobred]

Thsi is what I came up with
for [tex]x \geq ct[/tex]
[tex]u(x,t)=\dfrac{1}{6c}\ln\left[\dfrac{1+(x+ct)^{3}}{1+(x-ct)^{3}}\right][/tex]
and [tex]x < ct[/tex]
[tex]u(x,t)=\dfrac{1}{6c}\left[\ln\left(\dfrac{1+(x+ct)^{3}}{1+(ct-x)^{3}}\right)\right][/tex]