# How to solve this multi-variable non-linear differential equation?

This is a problem from my Classical Mechanics course, and it is: Suppose the potential in a problem of one degree of freedom is linearly dependent upon time, such that the Hamiltonian has the form

where $A$ is a constant. Solve the dynamical problem by means of the Hamiltonian-Jacobi equation, under the initial conditions $q=0$, $p=mv_0$ at $t=0$.

I substituted the Hamiltonian into the Hamilton-Jacobi's equation:

However I do not know how to solve for S.

• Possible duplicate: physics.stackexchange.com/q/109598/2451 Apr 19 '17 at 19:04
• Thank you Qmechanics. The answer you presented requires known solution to Hamiltonian, which seems to have defeated the purpose of the question, which is to use the H-J equation. Is there are way of solving for S if I do not know the solution to Hamiltonian? Apr 19 '17 at 19:23