In physics this variable is often the time, but it can also be the radial coordinate in problems with spherical symmetry.
ODEs can have a variety of forms as shown in the examples below.
However, it is possible to recast these equations in a regular form. The 2nd order equation for simple harmonic motion, can be rewritten in Hamiltonian form as two coupled 1st order equations, A solution is usually not defined without first establishing boundary values. If the solution is given at some initial time, we have an initial value problem. If the solution is given at an initial and final time, we have a two-point boundary value , or simply a boundary value problem. We shall concentrate on selected examples to focus our discussion.