Separation of variables

Math

Definition

A technique for solving differential equations by rearranging so that each variable and its differential appear on opposite sides of the equation. After separating, you integrate both sides independently.

How It Works

  1. Rewrite the differential equation so all terms involving y (and dy) are on one side and all terms involving x (and dx) are on the other.
  2. Integrate both sides independently.
  3. Solve for y if possible.
  4. Use an initial condition to find the constant of integration, if given.

Examples

  • Solving dy/dx = xy by rewriting as (1/y) dy = x dx and integrating both sides
  • Modeling population growth with dP/dt = kP, which separates to (1/P) dP = k dt

Related Terms

Antiderivatives Slope fields Exponential growth and decay

Study This Concept

Practice separation of variables with free review games in these units:

AP Calculus AB Unit 7: Differential Equations