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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.

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  • $\begingroup$ what was the general solution? $\endgroup$ – Mostafa Ayaz Jan 31 '18 at 8:13
  • $\begingroup$ Sorry i will correct the grammar mistake, the general solution is the one I wrote above $\endgroup$ – nan Jan 31 '18 at 8:15
  • $\begingroup$ Now substitute $f(ax+bt)$ in the PDE and proceed. This is D'Alembert method........ $\endgroup$ – Mostafa Ayaz Jan 31 '18 at 8:21
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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.

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