Linear Differential Equations of the First Order

A first-order linear differential equation is one in which $y$ and $\dfrac{dy}{dx}$ occur only to the first power and are not multiplied together — the standard form is

where $P$ and $Q$ are functions of $x$ alone (either may be constant).

Integrating factor (I.F.). Multiply through by

This is chosen precisely so that the left side becomes the exact derivative $\dfrac{d}{dx}\big(y \cdot \text{I.F.}\big)$. Integrating both sides then gives the general solution:

The method, step by step:

1. Write the equation in standard form and identify $P(x)$ and $Q(x)$.
2. Compute $\int P\,dx$ and then $\text{I.F.} = e^{\int P\,dx}$.
3. Write $y\cdot\text{I.F.} = \int Q\cdot\text{I.F.}\,dx + c$ and integrate the right side.

Worked illustration. Solve $\dfrac{dy}{dx} + \dfrac{y}{x} = x^2$.

Here $P = \dfrac1x$, $Q = x^2$. Then $\int P\,dx = \int \dfrac{dx}{x} = \ln x$, so $\text{I.F.} = e^{\ln x} = x$. Therefore

giving $xy = \dfrac{x^4}{4} + c$, i.e. $y = \dfrac{x^3}{4} + \dfrac{c}{x}$.