calculation – Equation of displacement in a vector field

When creating a program for a physics course at the university level, I have a vector field that represents a force such that $ vec F = F_x hat {i} + F_y hat {j} $. They ask me to create a program that shows the path of a mass $ m $ with an initial velocity vector ($ vec v_o $) and the position vector ($ vec x_o $) under the influence of the field.

My solution is to do the following steps:

$ vec F = m frac { partial ^ 2 vec x} { partial t ^ 2} $ and as such $ frac { neighbor F} {m} = frac { partial ^ 2 vec x} { partial t ^ 2} $

By integration of each side, I find that the path would be

$ vec x = frac { vec F Delta t ^ 2} {2m} + vec v_o Delta t + vec x_o $ Which is the same as the typical displacement equation.

My question is this; Is this intuitive use of correct calculation considering that $ vec F $ Does the position depend and, therefore, also depends on the time?