A first order differential equation is a differential equation that contains only first derivatives. In other words, it does not contain any derivatives higher than the first derivative, such as second derivatives, third derivatives, etc.
In Figure 1 below, you can see a third order differential equation. You can tell that it is a third order differential equation because it contains derivatives all the way up to order 3.
From left to right in the equation below, we have the third, second, and first derivatives of y(t), respectively. A first order differential equation would not contain the second and third derivatives. It would only contain the first derivative.
