http://www.mathworks.com/company/newsletters/news_notes/clevescorner/may03_cleve.html
In class, we’ve discussed some numerical methods for solving initial value problems (Euler’s method being one of them). One thing mentioned but unexplored in lecture was the concept of stiff differential equations. An equation is said to be stiff if some component of the equation varies slowly while another varies quite rapidly. The book (Cheney and Kincaid) offers the following example:

Here, the e^(-39t) terms seem inconsequential, i.e. if we were to use Euler’s method to evaluate x(5), we could do so with a relatively large step size and still maintain decent precision. The book goes on to explain, however, that a small h size is actually necessary for Euler’s method to converge. For trickier stiff equations, the process of finding an appropriate step size is hard.

Fortunately, MATLAB has some built-in functionality for solving stiff ODEs.  The article I’ve linked provides an example of a system of stiff equations that model a flame’s height on a match from t=strike to t=settle. The author goes through how to solve the system using MATLABs functions. If you try it yourself, you’ll see that the stiff equation solver works much faster than the normal equation solver.