First-Order Linear Differential Equations

A rigorous integrating-factor method for first-order linear equations, including intervals and verification.

Classify → choose method → apply condition. A differential equation is classified by order, linearity, and separability before a method is selected. The final condition determines constants and a residual check tests the solution.

Precise definition

A first-order equation is linear when it can be written $y'+P(x)y=Q(x)$. On an interval where $P$ and $Q$ are continuous, the integrating factor $\mu(x)=e^{\int P(x)dx}$ converts the left side to $(\mu y)'$, giving $\mu y=\int\mu Q\,dx+C$.

Notation and mathematical language

The coefficient of $y'$ must first be normalized to 1. Multiplying $\mu$ by a nonzero constant changes neither the method nor final family. The homogeneous solution is $Ce^{-\int Pdx}$; one particular solution accounts for $Q$.

Conceptual picture

The integrating factor is designed from the product rule: $(\mu y)'=\mu y'+\mu'y$. Requiring $\mu'=P\mu$ makes this exactly $\mu(y'+Py)$. The procedure therefore derives from a differential identity rather than a formula to memorize.

Fully worked example

Interpretation and application

Linear equations model decay with input, circuits, mixing, and first-order control. Superposition applies to the homogeneous operator, while the forcing $Q$ adds a particular response. Estimated coefficients make the model approximate even when its differential equation is solved exactly.