# Differential Equations

## What are Differential Equations?

Three hundred and fifty years ago, Isaac Newton wrote:

"It is useful to solve differential equations."

Since that time, researchers in many fields have found this statement to be true, particularly in recent times, with the advent of powerful personal computers making it possible to solve complicated differential equations numerically in a routine manner. Newton's enthusiasm was due to his success in using differential equations to give the first mathematical description of planetary orbits. Can you imagine what his reaction would be to the present widespread use of differential equations, with applications ranging from the space shuttle guidance system to epidemic models and neural networks?

Newton was referring to ordinary differential equations. The simplest example is:

where k is a constant. Think of y = y(t) as an unknown function that has to satisfy this equation. If you have studied calculus you may be able to guess the solution of this differential equation!

Now lets talk about partial differential equations. A simple example is:

where c is a constant. Here u = u(x,t) is an unknown function of position x and time t. This equation, which models various types of waves, for example the waves that travel along a vibrating guitar string, is called the wave equation.

Three of our foundational courses, AMATH 250, AMATH 351, and AMATH 353 will give you a comprehensive introduction to the important area of differential equations.