A scalar ordinary differential equation (ODE) is an equation that relates the derivative(s) of a single function of one variable to possibly the independent variable and the function itself. For example,

dy dt

(t) = ky(t)

is the ODE for exponential growth or decay that you saw in calculus. In this section and chapter, we will be most concerned with methods for solving ODEs.

At the end of Section 3.2, we will also present three basic “existential results” that give a firm foundation for all of the techniques we will learn. The order of the ODE is the highest derivative of that function in the equation. We will

study first-order ODEs that can be written in the form