Linear Differential Equations of the Type dy/dx + Py = Q

A first-order differential equation is called linear in $y$ if it can be written in the standard form

where $P$ and $Q$ are given functions of $x$ only (or constants) -- crucially, $y$ and $\dfrac{dy}{dx}$ each appear to the first power only, with no products such as $y\cdot\dfrac{dy}{dx}$ or $y^2$.

Why an integrating factor is needed. The left-hand side $\dfrac{dy}{dx} + Py$ is not, by itself, the derivative of any simple product -- so the equation cannot be integrated directly. The idea is to multiply the whole equation by a cleverly chosen function $\mu(x)$, called an integrating factor (I.F.), chosen so that $\mu\!\left(\dfrac{dy}{dx} + Py\right)$ becomes exactly $\dfrac{d}{dx}(y\mu)$ -- a single exact derivative that integrates immediately.

Deriving the integrating factor. By the product rule, $\dfrac{d}{dx}(y\mu) = \mu\dfrac{dy}{dx} + y\dfrac{d\mu}{dx}$. Comparing this with $\mu\dfrac{dy}{dx} + \mu P y$ (the left side after multiplying by $\mu$), the two match term-by-term exactly when

This is itself separable (Section 4): integrating both sides, $\ln|\mu| = \displaystyle\int P\,dx$ (taking the constant of integration as $0$, since only one valid $\mu$ is needed), so

General solution. With $\mu = e^{\int P\,dx}$, the equation $\mu\dfrac{dy}{dx} + \mu P y = \mu Q$ is, by construction, $\dfrac{d}{dx}(y\mu) = \mu Q$. Integrating both sides with respect to $x$:

$y\cdot\text{I.F.} = \int (Q\cdot\text{I.F.})\,dx + C,$ …