# PDE system and classical wave equation solution

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

Given a classical solution $$u(t,x)$$ to the wave equation, how do I construct a function $$v(t,x)$$ such that $$u(t,x)$$ and $$v(t,x)$$ form a solution to the ﬁrst-order system?

Someone who can help me with this question? I have no clue how to get to the answer.

If you have a solution $$u(x,t)=f(x+t)+g(x-t)$$ of the wave equation, then $$v_t=u_x=f'(x+t)+g'(x-t)$$ implies $$v(x,t)=f(x+t)-g(x-t)+C(x),$$ and the second equation tells you that $$C'(x)=0$$, that is, $$C$$ is a constant.