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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.

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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.

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