Help with PDE: F(t)g(r)+V/R Derivative

[matteo86bo]

I need help with this PDE, it's not an homework, I need to solve it for my thesis and it has physical application...anyway the problem is:
$$\frac{dx}{dt}=f(t)g(r)+\frac{v}{r}\frac{d (Rx)}{dR}$$

$$f(t)$$ and $$g(r)$$ are known.

I can solve the equation with only the first or the second term ...
actually I'm interest in how the second term modify the solution of the equation with the first term only. suggestions?

 

[kosovtsov]

Your PDE can be solved with help of Laplace transform method. For your purpose it'll be better the following form of general solution ( I assume that in fact R is r)

$$x(t,r) = \frac{1}{r}[\int_c^tf(\xi)g(vt-v\xi+r)(vt-v\xi+r)d\xi+F(vt+r)],$$

where $$F(z)$$ is an arbitrary function, $$c$$ is an arbitrary constant.