3. Motion in a nonlinear resistive medium.

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 




and thus the differential equation is transformed to 




Remark 1: The unperturbed problem 


has the solution


Remark 2: The perturbed problem has the solution


Hint: make the substitution z=1/y.