## Monday Math 3

General second order differential equation:

With independent variable t and unknown function y(t), our general form is. If this is nonlinear, there is no single general method of solving; however, there are two special sub-cases that allow reduction to first order equations (which in turn may or may not be solvable).

I. Position independent
This is the case where f has no explicit dependence on y:In this case, we define , and so the equation becomes , a first order differential equation in v(t). Solving for v(t), we then integrate with respect to time to find y(t).

II. Time independent
This is the case where f has no explicit dependence on t:In this case, we again define , and use the fact that, and our equation becomes;, which is a first order equation in v(y). Solving this, we get , which is itself a separable first order differential equation for y(t).

We can also consider some restricted subcases, which are even easier to solve:

III. Position and time independent
This is , or using v, , which is a separable first order equation: . One integrates that, then solves for v, and lastly integrates the result with respect to time to obtain the solution.

IV. Time and velocity independent
This is . As in case II, we use  and , to obtain, which is separable:, where F(y) is an antiderivative of f(y)As before, this is separable:

In the physics problem last Friday, our radial equation of motion was of form IV, and we used the method of defining  and using  to convert to a first order separable equation (though we made a few changes of variable before solving that equation). In fact, as most equations of motion are second order equations, these methods can be used in many problems, most notably those where the forces do not explicitly depend on time.