How can I simplify a PDE with variable coefficients?

[vargasjc]

I have spent hours and hours trying to solve a problem until reaching a dead point. I don't know how to solve, simplify, or proceed next with the following PDE:

$$m \frac{\partial^2 x}{\partial t^2} + 2 \frac{\partial m}{\partial t} \frac{\partial x}{\partial t} + x \frac{\partial^2 m}{\partial t^2}=0$$

Or, in other words: mx_tt + 2m_tx_t + xm_tt = 0

$$m x_{tt} + 2 m_{t} x_{t} + x m_{tt}$$

I thought it'd be easier but I don't seem to be able to simplify it. Maybe it is a form I don't recognize.

JC

 

[Office_Shredder]

What are you trying to solve for? x and m are both unknown functions?

 

[ross_tang]

Your equation can be written as:

$$\frac{\partial ^2(m x)}{\partial t^2}=0$$

And this can be solved easily. Hope it helps you.

 

[vargasjc]

Thanks.

It helped me solve the differential equation, and it also made me realize that the solution is not what I expected it to be. I believe I need to revisit the formulation.

Thanks again,

JC

 

[ross_tang]

You are welcome. Update us with your problem if you want.

 

[vargasjc]

I've ended up with a variable coefficient wave equation, so I'm browsing for numerical solutions of similar problems at ScienceDirect.

A closed form solution at this point is like asking Santa Claus for Dominion Over the Universe (yeah, um... not going to happen).

JC