Suppose that a body with mass m and initial velocity V0 is moving rectilinearly in a medium that affects the body with the motion resistance
where v=v(t), t>0, is the speed of the body at time t and a and b are positive constants with
Newton’s second law then gives that the equation of motion for the body is
Remark: If b=0 then
Scaling: Introduce the dimensionless variables
Then
and thus the differential equation is transformed to
where
Remark 1: The unperturbed problem
has the solution
Remark 2: The perturbed problem has the solution
Hint: make the substitution z=1/y.