consider the partial differential equation $u_{tt}=u_{xx}$
with boundary condition $u_x(0,t)=u_t(0,t)$
and initial conditions $u(x,0)=f(x)$ and $u_t(x,0)=0$
How to solve this with the method of characteristics, that is with knowledge that $u(x,t)=F(x-t)+G(x+t)$ is a solution to $u_{tt}=u_{xx}$ ? I know how to solve this with homogeneous boundary condition like dirichlet or neumann, but not with this type. With D'Alembert we find the following condition $u_t(0,t)=f'(x)/2+f'(-x)/2$. How to continue?