Separation of Variables Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Separation of Variables.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A method for solving first-order DEs of the form \frac{dy}{dx} = f(x) \cdot g(y): rearrange to \frac{dy}{g(y)} = f(x)\,dx, then integrate both sides.

If the rate of change factors into a piece that depends only on x and a piece that depends only on y, you can sort them onto opposite sides of the equation—all the y-stuff on the left, all the x-stuff on the right—then integrate each side in its own variable.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Separation of variables works only when \frac{dy}{dx} can be written as a product of a function of x alone and a function of y alone. It converts a DE into two independent integration problems.

Common stuck point: After integrating, you often get y defined implicitly (\ln|y| = \ldots). You may need to solve for y explicitly, and don't forget to handle \pm from absolute values.

Sense of Study hint: Try rewriting dy/dx as a product of something in x only times something in y only, then move all y-terms to the left side with dy.

Worked Examples

Example 1

easy
Solve dy/dx = xy with y(0)=2.

Solution

  1. 1
    Separate: dy/y = x\,dx. Integrate: \ln|y| = x^2/2+C.
  2. 2
    y = Ae^{x^2/2}. y(0)=2 \Rightarrow A=2.
  3. 3
    Solution: y = 2e^{x^2/2}.

Answer

y = 2e^{x^2/2}
Separate, integrate both sides, exponentiate, apply IC.

Example 2

hard
Solve the logistic DE dy/dt = y(1-y) with y(0) = 1/2.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Solve dy/dx = x/y with y(0)=3.

Example 2

medium
Solve Newton's cooling: dT/dt = -0.1(T-20), T(0)=80.
← Back to Separation of Variables Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

differential equations introintegral