Separable Differential Equations

Definition

An equation of the form: $\frac{dy}{dx} = g(x) H(y)$ is called separable.

• How to read: “The derivative of y with respect to x equals g of x times H of y.”
• Meaning: The derivative factors into a function of $x$ alone times a function of $y$ alone—variables can be split to opposite sides.

Why It Matters

Separable equations are the ‘low-hanging fruit’ of modeling; they allow us to solve complex, time-varying problems by breaking them into simpler parts, making them the primary tool for basic population dynamics and heat transfer calculations.

Core Concepts

A first-order differential equation is separable if it can be written as a product of a function of $x$ and a function of $y$.

• Solution Method To solve, “separate the variables” by moving all terms involving $y$ to one side and all terms involving $x$ to the other, then integrate: $\int \frac{1}{H(y)} \, dy = \int g(x) \, dx$
• How to read: “The integral of one divided by H of y dy equals the integral of g of x dx.”
• Meaning / when to use: Integrate both sides after separating. Solve for $y$ explicitly if possible, or leave as an implicit relation.

Connected Concepts

• Classification of Differential Equations
• Superposition Principle (Differential Equations)
• Autonomous Differential Equations