Solving differential equations using modified Picard iteration

Many classes of differential equation are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and engineering and include non-linear as well as linear differential equations. Examples involving partial as well as ordinary differential equations are presented. The method is easy to implement on a computer and the solutions so obtained are essentially power series. With its conceptual clarity (differential equations are integrated directly), its uniform methodology (the overall approach is the same in all cases) and its straightforward computer implementation (the integration and iteration procedures require only standard commercial software), the modified Picard methods offer obvious benefits for the teaching of differential equations as well as presenting a basic but flexible tool-kit for the solution process itself.