Linear First-Order Ordinary Differential Equations

A linear first-order differential equation is an ODE of the form:

\frac{d y}{d x} + P (x) y = Q (x)

where $P (x)$ and $Q (x)$ are continuous functions on a given interval $I$ .

The Integrating Factor Method

The standard technique for solving such equations involves finding an integrating factor, denoted by $μ (x)$ , which transforms the left-hand side of the equation into the derivative of a product.

Derivation of the Integrating Factor Formula

We seek a function $μ (x)$ such that:

μ (x) (\frac{d y}{d x} + P (x) y) = \frac{d}{d x} [μ (x) y]

Expanding the right-hand side using the product rule:

μ (x) \frac{d y}{d x} + μ (x) P (x) y = μ (x) \frac{d y}{d x} + \frac{d μ}{d x} y

This implies:

μ (x) P (x) = \frac{d μ}{d x} ⟹ \frac{d μ}{μ} = P (x) d x

Integrating both sides yields the integrating factor formula:

μ (x) = \exp (\int P (x) d x)

General Solution

Multiplying the original ODE by $μ (x)$ , we obtain:

\frac{d}{d x} [μ (x) y] = μ (x) Q (x)

Integrating with respect to $x$ :

μ (x) y = \int μ (x) Q (x) d x + C

Thus, the general solution is:

y (x) = \frac{1}{μ (x)} (\int μ (x) Q (x) d x + C)

Theoretical Context

The existence and uniqueness of solutions for this class of equations are guaranteed by the Picard--Lindelöf theorem provided $P (x)$ and $Q (x)$ are continuous. In the broader context of integrating factors, this specific formula represents the simplest case where the factor depends only on the independent variable.

For non-linear generalizations, one may refer to the Bernoulli equation or the Riccati equation.