Separable equations

An example of a separable differential equation

The Essentials

A separable differential equation is one where you can get all the x’s on one side and all of the y’s on the other. They fit in this form:

\[ \frac{dy}{dx} = f(x) g(y) \]

To solve a separable differential equation, divide through by g(y) and directly integrate:

\[ \int \frac{1}{g(y)} \, dy = \int f(x) \, dx \]

Example

Solve the equation:

\[ \frac{dy}{dx} = 8x^3y - 2y \]

First, fit it into the separable form:

\[ \frac{dy}{dx} = (y)(8x^3 - 2) \]

Then, separate and integrate:

\[ \frac{1}{y} \frac{dy}{dx} = 8x^3 - 2 \] \[ \int \frac{1}{y}dy = \int (8x^3 - 2) dx \] \[ \ln(y) + C_1 = 2x^4 + 2x + C_2 \]

Then, solve for y:

\[ \ln(y) = 2x^4 + 2x + C_3 \] \[ y(x) = e^{2x^4 + 2x + C_3} \] \[ y(x) = e^{2x^4} e^{2x} e^{C_3} \] \[ y(x) = C_4 e^{2x^4} e^{2x} \]

Practice

Evaluate this expressions:

\[ \frac{dy}{dx} = \frac{3x^2}{y - 2} \]

Solution:

\[ y(x) = Ce^{x^3} + 2 \]