Hi, the two forces acting on an object are its weight and air resistance. In my differential equations coursework I begin with 'air resistance = -kv' where v is the velocity and k is a constant. This is easy enough to solve.

Findgin auxiliary equation

Finding particular integral

General Solution

when

Therefore

when

So the equation is

I would appreciate if you could first check the correctness of the above working.

My main question is: how do you find the differential equation when the air resistance = -kv^2

This time the air resistance is dependent on the square of the speed rather than the speed. Newton's second law becomes:

I am looking to solve this for s as I did for the other equations.