Section 9.1 – Introduction to Differential Equations

What is a “Differential Equation”? It is an equation with a “differential” in it. Duh. Some examples:

Where are they used? In Physics, you learn about falling motion and the acceleration due to gravity, a constant acceleration downwards. Another factor is wind resistance. But this is not constant; it depends on the velocity of the object. This is an example where the change in velocity (acceleration) depends on the velocity at which it is traveling. There are lots of other uses in Physics, Engineering, City Planning and other occupations. Visualization of heat transfer in a pump casing, by solving the heat equation.

Notice both of those problems used the exponential function. Why do you think this is? When we solve things, we are going to need derivatives to cancel out original functions. This happens often with the exponential function because its derivative is itself. There are entire college classes that deal with Differential Equations, but we will only touch on the basics.