In this chapter we discussed differential equations. The focus on this general topic was quickly directed the case of one unknown function (except for Systems of two mutually coupled linear equations at the end). We devoted the greater part of our attention to linear differential equations of lower order.
As soon as one solution of a linear GDV is found, it suffices to determine all the solutions of the corresponding homogeneous equation. Because the solutions of a homogeneous linear differential equation form a vector space, it is even enough to find exactly as many linearly independent equations as the order of the ODE. In the case of linear differential equations with constant coefficients, there is a good solution method, of which we only treated cases of order #1# and #2#. The correspondence between two coupled linear differential equations of the first order and second order linear ODEs extends to higher orders, but we did not discus it.
In the cases of order #1# and order #2# we have formulated a uniqueness theorem (but we did not prove it). It states that there, under mild conditions, a unique specific solution exists with one initial condition for the first order and two initial conditions at the same point for the second order, one for the value of #y# and one for the value of #y'# at that point.
For an arbitrary linear first-order differential equation, we have shown how to determine the general solution by means of two integrations. For an arbitrary linear second-order ODE this has not been achieved. However, we have indicated methods that, given a solution of the corresponding homogeneous equation, find a second, linearly independent, solution, and, with this knowledge, finds a particular solution (by means of variation of constants). Thus, once a homogeneous solution is found, we can describe the general solution.
We also treated in greater detail some non-linear equations of first order and first degree. In particular, we discussed some basic strategies to understand the solutions without writing them down right away; the most important of these involves the direction field. We also showed how to work with an assumption (Ansatz) about the form of a solution of the ODE containing parameters (for instant a polynomial or a linear combination of some trigonometric functions), that , when substituted in the ODE, yields equations in terms of the parameters. Solutions of these parameter equations lead to solutions of the ODE. For particular first-order ODEs of first-degree, we also discussed techniques like separation of variables and integrating factors.
Thus, a number of basic methods were presented. This chapter is a good start for further study of not only ordinary differential equations, but also of partial differential equations, a vast subject in which systems of differential equations with several variables and multivariate functions appear. The coupled nonlinear equations were only a small example of this.
For the treatment of linear equations of higher order (as discussed: there is a good solution method in the case of constant coefficients), we recommend studying Linear algebra as a preparation.