Solve : $$u_{xx}+u_{xt}-20u_{tt} = 0 ,~~~~~ u(x,0) = \phi(x), u_t(x,0) = \psi(x)$$
Well, I looked through the answer and am stuck at one very important part: The answer assumed i know how to derive the general solution for this type of question: First, i factorize and get $$\left(\dfrac{\partial}{\partial x}+5\dfrac{\partial}{\partial t}\right)\left(\dfrac{\partial}{\partial x}-4\dfrac{\partial}{\partial t}\right)u = 0$$
And I am confused on how to derive that the general solution which is the following : $$u(x,t) = f(x+\frac{1}{4}t)+g(x-\frac{1}{5}t)$$
I looked through the proof of general solution of the wave equation and was unable to mimic it. Please help.
Assume that $$w=(\partial_x -4\partial_t)u.$$
Then solve
$$\left[\partial_x+5\partial_t\right]w=0$$
by the method of characteristics.
Then solve
$$w=\left[\partial_x -4\partial_t\right]u,$$
in which $w$ is now a known function.