DE-BOOK Integrating Factor

Chapter 4 Integrating Factor

Not every differential equation will politely separate its variables for us. For first-order linear equations, there’s another powerful tool: the integrating factor method. This method works by multiplying the entire equation by a carefully chosen function—an “integrating factor”—that transforms the left side into something much easier to handle: the derivative of a single product.

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Here’s the intuition: if you start with an equation like

\begin{equation*} \frac{dy}{dx} + P(x)y = Q(x), \end{equation*}

there’s a function \(\mu(x)\) you can multiply through by to make the left-hand side “collapse” into

\begin{equation*} \frac{d}{dx}\big[\mu(x) y\big] = \mu(x) Q(x). \end{equation*}

That small algebraic trick opens the door to solving the equation directly by integration.

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In this chapter, we’ll learn where this mysterious integrating factor comes from, how to find it every time, and how to use it to solve any first-order linear differential equation. By the end, you’ll have a systematic three-step process for tackling a huge class of problems.

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