Rewriting a differential equation so that a perfect derivative is formed, making integration trivial. Employs the use of a "helper function" called the ***integrating factor*** to create a [[Product Rule|product rule]] structure on one side of the equation. # Derivation Consider the non-separable differential equation below: $ \begin{align} \frac{dy}{dt}+p(t)\,y&=q(t)\\ y'+py&=q \end{align} $ To reach product rule structure, multiply both sides by a helper function $\mu$: $ \mu y'+\mu py=\mu q $ Notice that $\mu'=\mu p$ satisifies product rule structure: $ \mu y'+\mu py=\mu y'+\mu'y=\frac{d}{dt}(\mu y) $ so we first solve this using the [[Method of Separation|method of separation]]: $ \begin{align} \frac{d\mu}{dt}&=\mu p\\ \frac{1}{\mu}\frac{d\mu}{dt}&=p\\ \int\frac{1}{\mu}d\mu&=\int p\,dt\\ \ln|\mu|&=\int p\,dt\\ \mu &= e^{\int p\,dt} \end{align} $ Then we plug $\mu'=\mu p$ back into the original differential equation: $ \begin{align} \mu y'+\mu'y&=\mu q\\ \frac{d(\mu y)}{dt}&=\mu q\\ \int\frac{d(\mu y)}{dt}dt&=\int\mu q\,dt\\ \mu y&=\int\mu q\,dt \end{align} $ While we need a form of $q$ to fully solve the equation, we can rearrange for $y$, giving us the general solution to this differential equation: $ y(t)=\frac{\int\mu q\,dt}{\mu} $ ## General Procedure 1. Convert equation into the standard form used in the derivation. Identify $p(t)$ and $q(t)$, ensuring the appropriate sign is used. 2. Integrate $p(t)$ to find an appropriate integrating factor 3. Multiply the differential equation by the integrating factor and integrate both sides, which leads to the following form: $ \mu y = \int\mu q(t)\,dt $ 4. Complete the integration prcoess, apply initial conditions, and solve for $y$