Let
$\begin{align} u(x,t)&=F(x+ct)+G(x-ct),\\ u(x,0)&=f(x),\\ u_t(x,0)&=g(x). \end{align}$
How can I show that
$\begin{align} F(x)&=\frac12f(x)+\frac1{2c}\int_{x_0}^xg(s)\,ds+C,\\ G(x)&=\frac12f(x)-\frac1{2c}\int_{x_0}^xg(s)\,ds+C\text{ ?} \end{align}$