Let's say I have a vector force field as <x, y>, I.e force depends on the position.
Now I have a body of mass m, initial position S = <p1, p2> and initial velocity u = <u1, u2>. How can I write an equation for the trajectory for this particle with respect to time?
According to Newton's equation of motion, $$\frac{d^2}{dt^2} \bar{r} = \frac{1}{m} \bar{F}(\bar{r})$$
If you have two dimensions, say $\bar{r} = x \cdot \hat{e_x} + y \cdot \hat{e_y}$, then you get two coupled differential equations, $$\frac{d^2}{dt^2} x(t) - \frac{1}{m} F_x(x(t), y(t)) = 0$$ $$\frac{d^2}{dt^2} y(t) - \frac{1}{m} F_y(x(t), y(t)) = 0$$
Depending on the force function, you solve for these differential equations with the initial conditions.