Many physical phenomena can be modeled using the language of calculus. For example, observational evidence suggests that the temperature of a cup of tea (or some other liquid) in a room of constant temperature will cool over time at a rate proportional to the difference between the room temperature and the temperature of the tea.

In symbols, if $t$ is the time, $M$ is the room temperature, and $f(t)$ is the temperature of the tea at time $t$ then $f'(t) = k(M-f(t))$ where $k>0$ is a constant which will depend on the kind of tea (or more generally the kind of liquid) but not on the room temperature or the temperature of the tea. This is Newton's law of cooling and the equation that we just wrote down is an example of a differential equation. Ideally we would like to solve this equation, namely, find the function $f(t)$ that describes the temperature over time, though this often turns out to be impossible, in which case various approximation techniques must be used. The use and solution of differential equations is an important field of mathematics; here we see how to solve some simple but useful types of differential equation.

Informally, a differential equation is an equation in which one or more of the derivatives of some function appear. Typically, a scientific theory will produce a differential equation (or a system of differential equations) that describes or governs some physical process, but the theory will not produce the desired function or functions directly.

Recall from section 6.2 that when the variable is time the derivative of a function $y(t)$ is sometimes written as $\dot y$ instead of $y'$; this is quite common in the study of differential equations.

1. First Order Differential Equations

2. First Order Homogeneous Linear Equations

3. First Order Linear Equations

4. Approximation

5. Second Order Homogeneous Equations

6. Second Order Linear Equations

7. Second Order Linear Equations, take two