# solve twice differentiable function

#### ianchenmu

##### Member
Let $u:\mathbb{R}^n\rightarrow \mathbb{R}$ be a twice differentiable function. The 2-dimensional wave equation is
$\frac{\partial ^2u}{\partial x^2}=\frac{\partial ^2u}{\partial t^2}$, where $(x,t)$ are coordinates on $\mathbb{R}^2$. Prove that if $f,g:\mathbb{R}\rightarrow \mathbb{R}$ are twice di erentiable functions, then $u(x,t) = f(x-t) + g(x+t)$ solves the 2-dimensional wave equation. Use this fact, or another, to solve the Boundary-Value problem where
$u(s,0) = sin(s) + cos(s)$ , $u(0,s) = -sin(s) + cos(s)$ , $\forall s\in \mathbb{R}$.