Integrating Factor — Definition, Formula & Examples

An integrating factor is a function you multiply through a first-order linear differential equation to make the left side collapse into the derivative of a product, allowing you to solve by direct integration.

For a first-order linear ODE in standard form $\frac{dy}{dx} + P(x)\,y = Q(x)$ , the integrating factor is $\mu(x) = e^{\int P(x)\,dx}$ . Multiplying both sides by $\mu(x)$ transforms the left side into $\frac{d}{dx}[\mu(x)\,y]$ , reducing the problem to a single integration.

Key Formula

\mu(x) = e^{\int P(x)\,dx}

Where:

$\mu(x)$ = The integrating factor
$P(x)$ = The coefficient of y after writing the ODE in standard form dy/dx + P(x)y = Q(x)

How It Works

First, rewrite the equation in standard form

\frac{dy}{dx} + P(x)\,y = Q(x)

. Compute the integrating factor

\mu(x) = e^{\int P(x)\,dx}

. Multiply every term by

\mu(x)

; the left side becomes

\frac{d}{dx}[\mu(x)\,y]

exactly. Integrate both sides with respect to

x

and solve for

y

. The constant of integration is determined if an initial condition is given.

Worked Example

Problem: Solve the differential equation dy/dx + 2y = 6 with initial condition y(0) = 1.

Identify P(x) and Q(x): The equation is already in standard form with P(x) = 2 and Q(x) = 6.

\frac{dy}{dx} + 2y = 6

Compute the integrating factor: Integrate P(x) = 2 and exponentiate.

\mu(x) = e^{\int 2\,dx} = e^{2x}

Multiply through and integrate: Multiply both sides by e^(2x). The left side becomes the derivative of e^(2x) y. Integrate both sides.

\frac{d}{dx}\bigl[e^{2x}y\bigr] = 6e^{2x} \implies e^{2x}y = 3e^{2x} + C

Solve for y and apply the initial condition: Divide by e^(2x), then use y(0) = 1 to find C.

y = 3 + Ce^{-2x} \quad\Rightarrow\quad 1 = 3 + C \quad\Rightarrow\quad C = -2

Answer:

y = 3 - 2e^{-2x}

Why It Matters

The integrating factor method is typically the first general technique taught in a differential equations course and appears constantly in engineering and physics. Circuits (RC/RL), mixing problems, and Newton's law of cooling all produce first-order linear ODEs solved this way.

Common Mistakes

Mistake: Forgetting to divide by the leading coefficient before identifying P(x). For example, in 3(dy/dx) + 6y = 9, students set P(x) = 6 instead of first dividing everything by 3.

Correction: Always rewrite the equation so the coefficient of dy/dx is 1. Here, divide by 3 to get dy/dx + 2y = 3, giving P(x) = 2.