Euler's Method
In undergraduate math classes, you're taught all these algebraic methods for taking derivatives and integrals and solving differential equations. We didn't mention Laplace transforms, which are even more complicated ways of solving differential equations algebraically. Now, for the dirty secret about differential equations they don't tell you in school, unless you major in engineering: most differential equations you come across in real life have no analytical solution.
The good news is there have been numerical methods for avoiding messy algebra for hundreds of years, and with the invention of computers, these methods have become standard. Even when there is an analytical solution, numerical methods can be almost as accurate for practical purposes as the analytical method and take a fraction of the time to get a solution.
The idea of Euler's method is very simple:
- Start at the known point.
- Calculate the derivative at this point...
