MTH 222: Differential Equations

During Newton's time, people realized that fundamental principles seemed to govern natural phenomena and that these principles could be expressed using mathematical language. They discovered-invented the most important and extremely accurate mathematical models for mechanical systems. These models are what we now call differential equations.

The famous relation $F=ma$ is simply a differential equation. The acceleration is the rate of change of the velocity and the relation holds throughout the movement (or evolution) of the system. The challenge, then, is to find out how the system moves (or evolves) with time. (For example, one of the most impressive scientific advances occurred in astrophysics. The model was used to predict with astonishing, near-perfect, accuracy future positions of planets and asteroids months and even years in advance.)

The mathematical theory of differential equations became (and still is) a burgeoning field of research. On the applied side many equations were (and are) also used to model phenomena in economics, biology, population dynamics, spread of contagious diseases, physics, chemistry, etc. The daunting problem remained of solving the equations, which was often mathematically proven to be impossible in the sense of finding formulae; merely describing the qualitative behavior is a difficult challenge. The advent of computers has had a tremendous impact on both the theory and applications of differential equations.

In the course you will mainly use ideas and techniques of Calculus. Some of the beautiful concepts from Linear Algebra are also extremely useful for understanding and classifying the behavior of these systems. Some of the mathematical goals include predicting features of the future evolution of the system, studying the effect that a change in the initial conditions may have for the outcome (will there be a catastrophic difference?), examining periodicity, equilibrium, stability, and attractors.