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