A First Course in Applied Mathematics

CHAPTER 6

ORDINARY DIFFERENTIAL EQUATIONS

A great variety of natural phenomena can be modeled as continuous problems, those whose solutions are functions that depend on or vary with time. In general, a relationship between a function and its derivatives is known, and we have to find the function itself out of that relation. A relation of this type is what is called a differential equation: an equation that depends on a function and its derivatives.

It is here where the most fundamental theories and methods from calculus, analysis and linear algebra are crucial for establishing the solutions of such differential equations. When we are dealing with one-dimensional differential equations, calculus suffices, but when we move to solve n-dimensional differential equations, we will need some of the most elegant and useful tools of linear algebra for computing a solution. At the same time, some theorems from analysis (or advanced calculus) become essential for establishing existence, uniqueness and other properties of the solutions of differential equations. This is probably why differential equations are considered one area of applied mathematics: Calculus, analysis, and linear algebra are applied to the theory and methods of differential equations.

Once we leave the textbook or classroom examples, most differential equations are still theoretically solvable by analytical methods, but, impossible to solve in practice due to their complexity. At that point, we need to make use of numerical ...