Suppose a three body scenario with an artificial satellite, a planet (much massive than the satellite), and a sun (much massive than the planet). There is no other significant force on these three bodies. We know the positions and velocities of the planet and the satellite relative to sun at a given instant ($t_0$). We need the satellite to be moving in a given direction, relative to the sun, at a given future instant ($t_1$). Any approximations that are valid for a gravity assist situation can be made (please mention in the answer). The objective here is to find a $\Delta V$ vector that can be applied at $t_0$ to achieve the needed final direction of the satellite. To make it really a gravity assist, most of the force on the satellite from $t_0$ to $t_1$ has to be from the planet's gravity, not the sun or the satellite's thrusters.
Can the objective be achieved without numerically solving differential equations? I am looking for an explicit equation (or a set of equations) that can be solved to get the $\Delta V$.
Is there any other data that if available, allows the use of any such method?