Question

We have talked in class about second order differential equations. These equations often arise in applicationsof Newtons second law of motion. For example, supposeyis the displacement of a moving object with massm. Its reasonable to think of two types of time-independent forces acting on the object. One type - suchas gravity - depends only on positiony. The second type - such as atmospheric resistance or friction -may depend on position and velocityy′. (Forces that depend on velocity are called damping forces.) Picka physical system you are familiar with and describe how you can use your knowledge of the behavior ofthat system to guess the solutions to the differential equation describing the system of your choosing. Howwould you check whether your guess is indeed a valid solution? How would you check that it is a completesolution?

Answer 1

A familiar system is the vertical motion of a point mass (m), gravity acts downwards while the drag force acts against the motion, for a falling particle the Newton's equation would look like:

where the drag has been denoted as f(v), since gravity and friction try to nullify each other, the particle will acquire a uniform 'terminal speed' after a characteristic time. Before this, the speed would increase from zero since initially gravity would dominate over friction.

The exact solution will depend on the functional dependence of f(v), the usual procedure to check if a solution is valid is to plug it into the differential equation. Further, in order to qualify as a complete solution, it must satisfy all initial conditions, such as initial speed and position.