Consider an infinite string stretched taut on $x$ axis from $-\infty$ to $\infty$ . Let the string be drawn aside into a curve $y=f(x)$ and released, and assume that its subsequent motion is described by the wave equation.
Use
$$y(x, t) = F(x + at) + G(x − at)$$
to show that the string’s displacement is given by d’Alembert’s formula :
$$y(x, t) = \frac{f(x + at) + f(x − at)}{2}$$
I have this problem from Simmons Differential Equations book, but I have no idea how to solve it
Using
$$y(x,t) = F(x + at) + G(x - at)$$
We have that
$$\begin{align} \\ y(x,0) &= f(x) \\ &= F(x) + G(x) \ \ \ \ \ \ \ \ \ \ (1)\\ \end{align}$$
You should also have another condition that
$$\begin{align} \\ \partial_t y(x,0) &= 0 \\ &= aF'(x) - aG'(x) \\ \implies F'(x) &= G'(x) \\ \implies F(x) &= G(x) \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\\ \end{align}$$
Substituting $(2)$ into $(1)$
$$\begin{align} \\ \implies f(x) &= F(x) + F(x) \\ &= 2F(x) \\ \implies \frac{1}{2}f(x) &= F(x) \\ \end{align}$$
But we also have
$$\begin{align} \\ f(x) &= G(x) + G(x) \\ &= 2G(x) \\ \implies \frac{1}{2}f(x) &= G(x) \\ \end{align}$$
Hence
$$\begin{align} \\ y(x,t) &= F(x + at) + G(x - at) \\ &= \frac{1}{2}f(x + at) + \frac{1}{2}f(x - at) \\ &= \frac{f(x + at) + f(x - at)}{2} \\ \end{align}$$