Classical solutions to the wave equation

Question

Suppose $u(t,x)$ and $v(t,x)$ are $C^2$ functions defined on $\Bbb R^2$ that satisfy the first-order system of partial differential equations $u_t$ = $v_x$, $v_t$ = $u_x$.

Now I have to answer the following questions, but I have no clue how to start.

1: Show that both $u$ and $v$ are classical solutions to the wave equation $u_{tt}$ = $u_{xx}$.

2: Which result from multivariable calculus do you need to justify the conclusion?

Answer 1

1. Since the functions belong to the $C^2$ class, you have $$ u_{tt}=\frac{\partial}{\partial t} u_t= \frac{\partial}{\partial t} v_x = \frac{\partial}{\partial x} v_t=\frac{\partial}{\partial x} u_x=u_{xx}\iff u_{tt} = u_{xx} $$
2. The result in multivariable calculus used to justify the conclusion is the following one: if $f\in C^2(\Bbb R^n)$, then the first order partial derivatives to $f$ commute each other, i.e. $$ \frac{\partial}{\partial x_i}\left( \frac{\partial f(\boldsymbol{x})}{\partial x_j} \right) =\frac{\partial}{\partial x_j}\left(\frac{\partial f(\boldsymbol{x})}{\partial x_i} \right)\quad\forall i,j=1,\ldots,n,\:\:\: i\neq j,\:\: \boldsymbol{x}=(x_1,\ldots,x_n) $$