Given a first-order differential equation in the form:

$\frac{dy}{dx} = g(x)h(y)$

The steps for solving using separation of variables are:

1. Separate the Variables: Arrange the equation so that all $y$ terms (including $dy$ ) are on one side, and all $x$ terms (including $dx$ ) are on the other side:

$\frac{1}{h(y)} \, dy = g(x) \, dx$

2. Integrate Both Sides: Integrate both sides with respect to their respective variables:

$\int \frac{1}{h(y)} \, dy = \int g(x) \, dx$

3. Solve for $y$ : After integration, solve the resulting equation for $y$ if possible.
4. Include the Constant of Integration: When integrating, always include the constant of integration (often denoted as $C$).

Example 1: Simple Separable Differential Equation

Solve the differential equation:

$\frac{dy}{dx} = xy$

Step-by-Step Solution:

5. Separate the Variables: Move all $y$ -terms to one side and $x$ -terms to the other:

$\frac{1}{y} \, dy = x \, dx$

6. Integrate Both Sides: Integrate the left side with respect to $y$ and the right side with respect to $x$ :

$\int \frac{1}{y} \, dy = \int x \, dx$

The integrals are:

$\ln |y| = \frac{x^2}{2} + C$

Where $C$ is the constant of integration.

7. Solve for $y$ : To solve for $y$, exponentiate both sides to eliminate the logarithm:

$y = e^{\frac{x^2}{2} + C} = e^C \cdot e^{\frac{x^2}{2}}$

Let $C_1 = e^C$ (since $e^C$ is just another constant):

$y = C_1 e^{\frac{x^2}{2}}$

This is the general solution.

Example 2: More Complex Separable Differential Equation

Solve the differential equation:

$\frac{dy}{dx} = \frac{x}{y + 1}$

Step-by-Step Solution:

8. Separate the Variables: Multiply both sides by $y + 1$ and $dx$ to separate the variables:

$(y + 1) \, dy = x \, dx$

9. Integrate Both Sides: Integrate the left side with respect to $y$ and the right side with respect to $x$ :

$\int (y + 1) \, dy = \int x \, dx$

The integrals are:

$\frac{y^2}{2} + y = \frac{x^2}{2} + C$

10. Solve for $y$: This equation is the implicit form of the general solution. Depending on the problem, you might leave it in this form or solve for $y$ explicitly.

The method of separation of variables is a powerful tool for solving first-order differential equations that can be separated into functions of $x$ and $y$ . By isolating the variables and integrating both sides, you can find the general solution to the equation, which usually includes an arbitrary constant. This method is particularly useful for a wide range of physical and mathematical problems, such as those involving growth and decay, motion, and fluid flow.