Find the solution of the wave equation using d'Alembert solution.
$u(0,t)=0$, $u(x,0)=0$
$u_t(x,0)=\frac{x^2}{1+x^3}, \, x\geq0$
$u_t(x,0)=0, \, x<0$
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 $u(x,0)=a(x)=0$
$u(x,t)=\frac{1}{2c}\left( \int^{x+ct}_{x-ct} dy\, b(y) \right)$
Where
$b(y)=\frac{y^2}{1+y^3}$
Is this correct?