Separation Of Variables

Level 1 - Calculus topic page in Differential Equations.

Principle

Separation of variables solves differential equations where all terms involving \(y\) can be moved to one side and all terms involving \(x\) to the other.

Notation

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

separable differential equation form

\(dy,dx\)

differentials used to separate variables

\(C\)

constant of integration

The Core Method

Rewrite the equation so each side contains only one variable, then integrate.

Separated form

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

After integrating, use any initial condition to determine the constant. Check for constant solutions that may be lost by division.

Worked Cases

Question

Solve \(\frac{dy}{dx}=2y\) for \(y\ne0\).

Answer

Separate variables: \(\frac{1}{y}dy=2dx\). Integrate both sides: \(\ln|y|=2x+C\). Exponentiate: \(|y|=e^Ce^{2x}\). Absorb constants to get \(y=Ce^{2x}\).

Examples

Question

Solve \(dy/dx=xy\) with \(y(0)=3\).

Answer

Separate: \(dy/y=x\,dx\). Integrate: \(\ln|y|=x^2/2+C\), so \(y=Ce^{x^2/2}\). Use \(y(0)=3\): \(C=3\). Therefore \(y=3e^{x^2/2}\).

Series Solutions