Application to Non-Linear Differential Equations

The linear-differential equations in physics can often be solved analytically whereas most non-linear ones can only be solved numerically. It is important therefore to be able to apply the ideas developed here to such cases. Consider the simple example

$${\d y\over\d t} + \alpha y^2 = 0.$$ (1.23)

In this case $f(y,t) = \alpha y^2$ and $\partial f/\partial y = 2\alpha y$ which can be substituted into (1.21) to give the stability condition

$$\delta t \le 1/(\alpha y),$$ (1.24)

which depends on $y$, unlike the simpler cases. In writing a program to solve such an equation it may therefore be necessary to monitor the value of the solution, $y$, and adjust $\delta t$ as necessary to maintain stability.