# D'Lembert method application

#### Markov

##### Member
Solve

\begin{aligned} & {{u}_{tt}}={{u}_{xx}},\text{ }x>0,\text{ }t>0 \\ & u(0,t)=0,\text{ }t>0 \\ & u(x,0)=x{{e}^{-{{x}^{2}}}},\text{ }0<x<\infty \\ & {{u}_{t}}(x,0)=0. \end{aligned}

The condition $u(0,t)$ is new to me, since I usually apply the method when only having $u(x,0)$ and $u_t(x,0),$ what to do in this case?

#### HallsofIvy

##### Well-known member
MHB Math Helper
The boundary and intial value conditions match at (0, 0) so I would just ignore it, then check to make sure my result satisfied that.

#### Markov

##### Member
Okay, I'll apply it then and see how it works, thanks!