Summary and Conclusions

At the center of science lies the "scientific method" in which real world problems are first translated into mathematical models, then the models are subjected to mathematical analysis and prediction, and finally the conclusions drawn from the model are tested in the laboratory and compared with the original real world problem. Normally the model does not exactly reflect the real state of affairs and the process is repeated by refining the mathematical model, making new predictions, and subjecting them to further testing in the laboratory. Over time the models improve.

Mathematical models very commonly take the form of differential equations. There is thus a great deal of interest in methods for solving differential equations. Sometimes the solution and analysis of the equation can be handled by algebraic methods involving exact formulas. In most cases, however, the equations will be too complicated. Then, numerical and graphical methods come to the rescue. Euler's method is a simple, yet powerful, technique for approximating solutions numerically and graphically in cases that do not yield to standard mathematical methods.