D'alembert solution for the semi-infinite string

[bobred]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

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

$$b(y)=<b>\frac{y^2}{1+y^3}</b>$$

Is this right?

 

[bobred]

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