---
title: "First-Order Linear Equations and the Integrating Factor"
description: "Multiply through by a factor chosen to turn the left side into the derivative of a product, then integrate once. The factor is built from the coefficient of y."
canonical: https://lightmysky.com/learn/mathematics/first-order-linear-equations-and-the-integrating-factor-mt_BXmt2pbWp8
source: https://lightmysky.com/learn/mathematics/first-order-linear-equations-and-the-integrating-factor-mt_BXmt2pbWp8.md
retrieved: 2026-09-12
---

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# First-Order Linear Equations and the Integrating Factor

Multiply through by a factor chosen to turn the left side into the derivative of a product, then integrate once. The factor is built from the coefficient of y.

Subject: Mathematics · Area: Differential Equations · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/first-order-linear-equations-and-the-integrating-factor-mt_BXmt2pbWp8

## Ready when they can

- Write an equation in standard form and compute its integrating factor
- Recognise the product rule pattern the factor creates
- Solve a mixing or cooling problem set up as a linear equation

## Lesson: The factor that builds a product

A first order linear equation always fits one template: y prime + P(x) y = Q(x). First divide so y prime has coefficient 1: 2y prime + 4y = 6 becomes y prime + 2y = 3. Now P is visible and the integrating factor is e raised to the integral of P dx. For P = 2 the factor is e to the 2x, and for y prime + (2 over x) y = 1 the factor is x squared, worth 9 at x = 3.

**Example.** Solve y prime + 2y = e to the x. Multiply everything by e to the 2x and the left side collapses into the derivative of e to the 2x times y. So (e to the 2x times y) prime = e to the 3x. Integrate once: e to the 2x times y = e to the 3x over 3 + C. Divide out: y = e to the x over 3 + C e to the minus 2x. Check it by substitution: y prime + 2y returns e to the x.

Mixing tanks and cooling bodies run on the same template, with inflow and decay playing Q and P. Take dy/dt + 0.5 y = 2. Over a long time the rate settles to zero, so 0.5 y = 2 and y drifts toward 4. You can read such steady states straight from P and Q.

**Tip.** Finish every problem the same way: divide first, multiply by the factor, collapse, integrate once, keep C. The classic stall is stopping right after the clever multiplication. And always plug your answer back into the equation to check.

**Recap.** Divide into standard form, multiply by e to the integral of P, collapse, integrate, and check.

## Practice

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

## Needs first

- [Separable Equations](https://lightmysky.com/learn/mathematics/separable-equations-mt_itSTsLTkwQ)
- [The Product and Quotient Rules](https://lightmysky.com/learn/mathematics/the-product-and-quotient-rules-mt_NJ7INJ0jfv)

## Opens up

- [Green's Functions for Boundary Value Problems](https://lightmysky.com/learn/mathematics/greens-functions-for-boundary-value-problems-mt_2ctFrfKIgW)
- [The Logistic Equation and Saturating Growth](https://lightmysky.com/learn/mathematics/the-logistic-equation-and-saturating-growth-mt_7Y_48Ga33L)