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I have the following problem:

$ \rho_{tt} +a\rho_{xt}-c^2\rho_{xx}=bv_{xx}$, where

$\rho=\rho(x,t)$, $v=v(x,t)$ and $a,b,c$ are constant.

My attempt to solve such an equation is to treat $v$ as any other function and just solve by meaning of change of variables (because the left part is a hyperbolic equation) and then look for a particular solution.

Is it correct?

Is there another possible approach to the problem?

When I try Fourier transform it gives me a second order non-homogeneous ODE which I don't get how to guess a possible particular solution.

Help is appreciated.

Ankara
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  • So, basically, $v(x, t)$ is given and we have to solve for $\rho$? – Robert Lewis Mar 21 '15 at 18:06
  • $v$ represents the Eulerian velocity of a fluid. Is it supposed to be given? – Ankara Mar 21 '15 at 18:16
  • Maybe, maybe not. As it stands, you've got one equation for the two functions $\rho$ and $v$, so unless there is another equation relating $v$ and $\rho$, you'll have to assume $v$ is known to solve $\rho_{tt} + a\rho_{xt} - c^2 \rho_{xx} = bv_{xx}$. – Robert Lewis Mar 21 '15 at 18:40

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