---
title: "Separable Equations"
description: "When the equation factors into a part in y and a part in x, separate the variables and integrate both sides. The constant of integration is where the initial condition enters."
canonical: https://lightmysky.com/learn/mathematics/separable-equations-mt_itSTsLTkwQ
source: https://lightmysky.com/learn/mathematics/separable-equations-mt_itSTsLTkwQ.md
retrieved: 2026-09-12
---

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# Separable Equations

When the equation factors into a part in y and a part in x, separate the variables and integrate both sides. The constant of integration is where the initial condition enters.

Subject: Mathematics · Area: Differential Equations · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/separable-equations-mt_itSTsLTkwQ

## Ready when they can

- Separate and integrate an equation, keeping the constant
- Apply an initial condition and state the interval where the solution is valid
- Recognise an equation that cannot be separated

## Lesson: Split, integrate, fix the constant

A separable equation splits into a y-piece times an x-piece: dy/dx = f(x) times g(y), perhaps after a little algebra. You park every y-piece with dy on one side and every x-piece with dx on the other. Then you integrate each side on its own. One constant of integration ties the two halves into a whole family of curves.

**Example.** Take dy/dx = 2x with y(0) = 3. Integrate: y = x squared + C. The condition fixes C: 3 = 0 + C, so C = 3. At x = 1 you get y = 1 + 3 = 4. Keep C from the start and the condition simply names it.

The condition selects one curve, and you must say where it stays valid. Dividing by y-expressions can hide solutions: dy/dx = x y is also solved by y = 0. The nonzero family is y = C e to the x squared over 2, and y(0) = 2 gives C = 2, valid for all x. With dy/dx = 2y over x for x greater than 0, you get ln|y| = 2 ln|x| + C, so y = C x squared. Then y(1) = 1 gives C = 1 and y(3) = 9, trusted only for x greater than 0 because you divided by x.

Test the shape first: products and quotients of one x-piece and one y-piece split, while sums like x + y never split. Never drop C: the condition fixes it, it does not replace it. Always name the interval where your divisions stay legal.

**Recap.** Separate the variables, integrate both sides, let the condition fix C, and state the interval.

## Practice

13 questions on this page, each with its working shown.

## Needs first

- [Direction Fields and Euler's Method](https://lightmysky.com/learn/mathematics/direction-fields-and-eulers-method-mt_Ak-TZOX3GR)
- [Integration by Substitution](https://lightmysky.com/learn/mathematics/integration-by-substitution-mt_jH-j_XsdC7)

## Opens up

- [First-Order Linear Equations and the Integrating Factor](https://lightmysky.com/learn/mathematics/first-order-linear-equations-and-the-integrating-factor-mt_BXmt2pbWp8)
- [Hydrostatic Equilibrium and the Structure of a Star](https://lightmysky.com/learn/science/hydrostatic-equilibrium-and-the-structure-of-a-star-mt_RKeMW8iX56)