Show that $u$ and $v$ are solutions of the wave equation Suppose that $u = u(x, t)$ and $v = v(x, t)$ have partial derivatives as follows:
$$u_t=-v_x \quad\text{and}\quad v_t=-u_x.$$
Show that $u$ and $v$ are solutions of the wave equation:
$$u_{tt}=u_{xx}.$$

My solution effort:
taking derivative of $v_t=-u_x$ w.r.t $x$, $v_{xt}=-u_{xx}$, we get $u_{xx}=-v_{xt}$. Similiarly; $u_{tt}=-v_{xt}$. Then $u_{tt}=u_{xx}$.

Is this right?
If $v$ is a $C^2$ function, then $v_{xt}=v_{tx}$ (Schwaz's theorem) and you are right.
$$u_{tt} =-v_{xt} =-(v_t )_x =-(-u_x)_x =u_{xx}  $$