Separable equations

Japanese school year: University year 2

What you learn

Learn how to solve separable differential equations by separating the variables onto opposite sides and integrating. This method is frequently applied to model physical phenomena like population growth and radioactive decay. Before starting, you should have a solid grasp of basic derivative rules and indefinite integration.

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Key points

This is a separable differential equation, where the derivative $dy/dx$ factors into a function of $x$ , $f(x)$ , times a function of $y$ , $g(y)$ . It is one of the most fundamental types of ODEs.

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

This formula solves a separable equation by separating variables: $y$ on the left and $x$ on the right, then integrating both sides (where $C$ is an integration constant).

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

Before dividing by $g(y)$ , check whether any constant value $y = y_0$ satisfying $g(y_0) = 0$ is a solution. Division by zero is undefined, so constant solutions must be identified separately.

g(y_0) = 0 \implies y = y_0

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Practice 1Coming soon