Higher Order Equations; Stability and Convergence

In this lecture we saw how differential equations of order two and higher can be written as a system of first order differential equations. The solution of such a linear system is entirely determined by the eigenvalues and eigenvectors of the matrix of the system.

We defined the stability of the solution to an initial value problem, investigating the sensitivity of the solution to changes in the initial values.

If we have stability, then we can solve the initial value problem numerically, and then we are concerned about the size of the step so the approximations converge to zero as we step further. We saw that requiring convergence leads to rather tight upper bounds on h when Euler method is used. The analysis of the modified Euler method was left as an exercise, but we concluded that for so-called stiff problems, implicit methods like the modified Euler method must be used.