A differential equation is an equation that relates the rate $\displaystyle\frac{dy}{dt}$ at which a quantity $y$ is changing (or sometimes a higher derivative) to some function $f(t,y)$ of that quantity and time.

Examples: $$\displaystyle\frac{dy}{dt} = 3y; \qquad \frac{dy}{dt}= 5t^2;\qquad \frac{dy}{dt} = 5t^2 + 3y $$ are examples of explicit first-order equations, i.e., equations of the form $$ \frac{dy}{dt} = f(t,y) $$ while $$ \frac{d^2y}{dt^2} = -4 x;\qquad \frac{d^2y}{dt^2}=y\sin(t)+\frac{dy}{dt} $$ are examples of explicit second-order equations, i.e., equations of the form $$ \frac{d^2y}{dt^2} = f\left(t,y, y'\right).
$$ In this class we will only be considering first-order equations.

Modeling with differential equations boils down to four steps.

  1. Understand the mechanics behind what we're trying to model. For example, if we are studying an ecomomic system, we need to know something about what might cause the changes in this economics system to occur.

  2. Express the rules for how the system changes in mathematical form. The result is a differential equation.

  3. Use calculus to solve the differential equation.

  4. Interpret the solution(s) in context to predict future behavior.

The following video describes this general philosophy. It is part of a series of videos originally made for M408D, and some of the things that it says we'll see later (like second-order equations) won't actually appear in M408R.