How can I solve the Hamilton-Jacobi equation for this time-dependent potential?

[dipole]

Homework Statement

I'm given the time-dependent potential,

$$V(x,t) = -mAxe^{-\gamma t}$$

and asked to find the solution to the Hamilton-Jacobi equation,

$$H(x,\frac{\partial S}{\partial x}) + \frac{ \partial S}{\partial t} = 0$$

The Attempt at a Solution

Without any additional information, I'm assuming the correct Hamiltonian is given simply by,

$$H = \frac{p^2}{2m} -mAxe^{-\gamma t}$$

which gives me,

$$\frac{1}{2m}\bigg ( \frac{\partial S}{\partial x} \bigg )^2 - mAxe^{-\gamma t} + \frac{ \partial S}{\partial t} = 0$$

but I'm having troule separating the variables in order to solve this equation. Normally, when $V = V(x)$ you can use the form $S(x,\alpha,t) = W(x,\alpha) - Et$, but here this won't work.

Have I somehow used the wrong Hamiltonian, or do I just need to guess correctly the right form of $S$?

 

[minimax]

I suggest using the method of characteristics to solve this PDE problem.