What remains to be taught is how to give the result broad applicability. Reducing a second order equation has already been treated in the introduction to the second unit. Extending this to higher order equations is almost an afterthought. Thus a problem which is already autonomous, but which has initial conditions given away from 0 can also be addressed in this manner. The problem of finding a system which maintains a solution but has a polynomial generator seems to be much more difficult, but actually turns out to be a nice arena in which to teach some applied algebra.
Polynomials simply stay as they are.
The sum, product, and composition theorems for derivatives all have polynomial forms that can be written with their terms, factors, or components. Thus the following list suffices to handle the problems in Boyce and DiPrima. This list can be taught since it is all covered in a beginning calculus course. Numerically implementing the process and getting to the screen - Applying Picard iteration involves being able to integrate, beginning with a constant function. As mentioned above, if the MacLaurin Series is the desired result, then the computation is also tractable numerically 3 .