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Chapter 6 Integrating Factor
📝: 🎧 Listen.
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 a single derivative.
Here’s the intuition: Every equation of the form
\begin{equation*}
\frac{dy}{dx} + P(x)y = Q(x)
\end{equation*}
can be multiplied by a special factor to give you
\begin{equation*}
\boxed{\phantom{M}}\cdot \frac{dy}{dx} + \ \boxed{\phantom{M}}\cdot P(x)y = \ \boxed{\phantom{M}}\cdot Q(x)
\end{equation*}
and the whole left side bcomes a single derivative:
\begin{equation*}
\frac{d}{dx}\big[\ \boxed{\phantom{M}}\cdot y\big] =\ \boxed{\phantom{M}}\cdot Q(x).
\end{equation*}
The goal of this chapter, is to learn where this mysterious factor comes from, how to find it, and how to use it to solve any first-order linear differential equation.
🗝️ Key Takeaways...
The integrating factor method solves any first-order linear differential equations of the standard form
\begin{equation*}
y' + P(x)y = Q(x)\text{.}
\end{equation*}
The integrating factor, \(\mu(x)\text{,}\) is the function we multiply onto the standard form to “complete the product rule” (on the left).
Completing the product rule is conceptually similar to completing the square; both introduce a missing piece that makes the equation easier to solve.
Multiplying by the integrating factor allows the left-hand side to be written as a single derivative, which can be easily solved by integration.
Once in standard form, \(\ y' + P(x)y = Q(x)\ \text{,}\) the method involves:
Computing the integrating factor \(\mu = e^{\int P(x)\, dx}\text{,}\)
Multiplying the standard form by
\(\mu\) to rewrite it as
\begin{equation*}
\frac{d}{dx}\left[\mu\cdot y\right] = \mu\cdot Q(x),\quad \text{and}
\end{equation*}
Applying direct integration.
