Separable Differential Equations

A separable equation is a first-order differential equation that can be split into two independent parts—one containing only x and the other only y.

The y terms are placed on one side and the x terms on the other, allowing separate integration.

For example, consider a hot cup of tea cooling in a room.

The rate at which the tea cools is proportional to the difference between its temperature and the room temperature. A negative sign is added to the proportionality constant k to indicate that the temperature decreases with time.

The equation is separable because it can be rewritten with temperature terms on one side and time on the other. Integrating both sides gives a logarithmic equation with a constant of integration.

When both sides are exponentiated, the general solution emerges, with a constant that may be positive, negative, or zero. The constant is found by substituting t equals zero and the initial temperature.

The temperature difference decays exponentially, causing the tea to cool quickly at first and then more slowly over time.