In Chapters ??, ??, and ?? we developed analytic methods for finding solutions to linear differential equations with constant coefficients and with forcing. In this chapter we show how to find closed form solutions to certain types of nonlinear equations as well as to certain linear equations with time dependent coefficients.
Most nonlinear differential equations cannot be solved analytically and for these equations one has to rely either on advanced theory or on numerical methods. There are, however, specific types of ordinary differential equations whose solution by hand is possible. In this chapter we describe several of these types. In Sections ?? and ?? we solve forced nonconstant coefficient linear differential equations and systems using variation of parameters. In Section ?? we show how to solve higher order equations by reduction to first order systems and by a variant of variation of parameters called reduction of order. Sometimes it is possible to transform a nonlinear equation into an equation that can be treated by one of the methods mentioned above. This is accomplished by substituting another function for the function in a suitable way. Simplifications by substitution are discussed in Section ??.
The chapter ends with a discussion of two types of differential equations whose solutions lie on level curves of a real-valued function: nonautonomous exact differential equations and autonomous Hamiltonian systems. Exact equations are treated in Section ??. Hamiltonian systems, which arise naturally in mechanical systems, are treated in Section ??.