Separation of variables is a technique used to solve certain types of ordinary differential equations (ODEs), particularly those that are first-order and separable. The method involves rearranging the terms of the differential equation to express variables on one side and constants on the other, allowing the equation to be integrated straightforwardly.

The general form of a first-order separable differential equation is: dy/dx = g(x)h(y)

Here, g(x) is a function of x and h(y) is a function of y. The goal of separation of variables is to rearrange the equation to isolate the variables on one side and constants on the other side, making it possible to integrate both sides.

Steps for solving a separable differential equation:

Separate Variables: Express the differential equation in a form where all terms involving y are on one side and all terms involving x are on the other side.

Integrate Both Sides: Integrate both sides of the equation with respect to their respective variables.

Include the Constant of Integration: Don't forget to include the constant of integration on the side involving y.

Solve for y: If possible, solve the equation for y to obtain the explicit form of the solution.

The solution obtained from separation of variables is often expressed implicitly, and additional information (initial or boundary conditions) may be needed to determine the constant of integration.

Example:

For dy/dx = x^2y, the solution involves the steps of separating variables, integrating, including the constant of integration, and solving for y.

Separation of variables is a powerful technique that is particularly useful for solving certain types of differential equations encountered in various fields, including physics, engineering, and biology.