# Definition

A separable differential equation is one of the simplest types of differential equations. In basic terms, it is a differential equation where you can put the x’s on one side, and y’s on the other.

# Solving

Example 1:

## Example 2: Newton’s Law of Cooling

If you have a hot cup of coffee, it cools really quickly in the beginning. As the temperature of the coffee gets closer to room temperature, it starts cooling slower. Once the coffee is almost room temperature, it basically stays constant. If the coffee starts off at room temperature, it doesn’t cool at all. This is called **Newton’s Law of Cooling.** The rate of the temperature change is proportional to the difference of the body’s temperature and the surrounding’s temperature.

**Missing Solution**

Without the condition that T is greater than M, we would be missing a solution. If T = M, then the constant D would be 0. Why did we get this issue? We divided by T-M in the beginning, which only works if that’s non-zero.

**Practice Problem**

## Example 3: Radioactive Decay

For this case, the rate at which a radioactive isotope decays depends on the current amount of that isotope.

So, y’ = ky

**Practice Problem**

Pu-239's half life is 24,100 years. Now, suppose 10 grams of Pu-239 was released. How long will it take for the 10 grams to decay to 1 gram?