If F and G have second partial derivatives, show that $U(x,t)=F(x+at)+G(x-at)$ satisfies the wave equation: $$a^{2}\frac{\partial^2U}{\partial x^2}=\frac{\partial^2 U}{\partial t^2}$$. Sol: $$\\$$ I first say that $s=x+at$ and $w=x-at$, then i try to use the chain rule $$\frac{\partial U}{\partial t}=\frac{\partial U}{\partial s}\frac{\partial s}{\partial t}+ \frac{\partial U}{\partial w}\frac{\partial w}{\partial t}$$ $$\frac{\partial U}{\partial t}=\frac{\partial (F+G)}{\partial s}\frac{\partial s}{\partial t}+ \frac{\partial (F+G)}{\partial w}\frac{\partial w}{\partial t}$$ For propierties: $$\frac{\partial U}{\partial t}=\left( \frac{\partial F}{\partial s}+\frac{\partial G}{\partial s} \right) \frac{\partial s}{\partial t}+ \left( \frac{\partial F}{\partial w}+\frac{\partial G}{\partial w} \right)\frac{\partial w}{\partial t}$$ $$\frac{\partial U}{\partial t}=a\left( \frac{\partial F}{\partial s}+\frac{\partial G}{\partial s} \right) - a\left( \frac{\partial F}{\partial w}+\frac{\partial G}{\partial w} \right)$$ $$\frac{\partial^2 U}{\partial t^2}=a\frac{\partial}{\partial t}\left( \frac{\partial F}{\partial s}+\frac{\partial G}{\partial s} \right) - a\frac{\partial}{\partial t}\left( \frac{\partial F}{\partial w}+\frac{\partial G}{\partial w} \right)$$ $$\frac{\partial^2 U}{\partial t^2}=a\left( \frac{\partial F}{\partial t\partial s}+\frac{\partial G}{\partial t\partial s} \right) - a\left( \frac{\partial F}{\partial t\partial w}+\frac{\partial G}{\partial t\partial w} \right)$$
I do not know what to do in this part. I appreciate help.
Good start. It looks like you're having some bookkeeping issues. I think in this case it may be easier to not wrap $x+at$ and $x-at$ as separate variables. Note that $F$ and $G$ are both functions of one variable, the mixing of the variables happens in the $x\pm at$ arguments. Let's see how it goes.
\begin{align*} \frac{\partial}{\partial t} U(x,t) &= F'(x+at)\frac{\partial}{\partial t}(x+at) + G'(x-at)\frac{\partial}{\partial t}(x-at)\\ &= aF'(x+at) - aG'(x-at) \end{align*} and \begin{align*} \frac{\partial}{\partial x}U(x,t) &= F'(x+at)\frac{\partial}{\partial x}(x+at) + G'(x-at)\frac{\partial}{\partial x}(x-at)\\ &= F'(x+at) + G'(x-at). \end{align*} Can you take the next set of derivatives from here?