Separable First-Order of Differential Equation

A first-order differential equation is said to be separable if , after solving for the the derivative,

 the right-hand side can be factored by function of x and function of y,

If the factoring is not possible, this means that the derivative is not separable.

Let's have an example,

First, we check if it is separable,

Since it is separable, we can continue solving it by integrating both sides:

Another application for this is with Lotka-Voltera model of predator y - prey x dynamics.
It is given by,

 Where,

 Independently, they can't be solve in linear method. However, the dependence of the predator prey population can be solve by,

 This now becomes a separable fode as proof below,

After we integrate both sides, we are left with an implicit function.