First-Order Linear Differential Equations

Definition

A first-order linear differential equation is an equation that can be written in the standard form: $\frac{dy}{dx} + P(x)y = Q(x)$ where the dependent variable $y$ and its derivative $dy/dx$ appear only to the first power and are not multiplied together.

• How to read: “The derivative of y with respect to x, plus P of x times y, equals Q of x.”
• Meaning: Canonical first-order linear ODE — linear in $y$ and $y'$, with coefficient $P(x)$ and forcing term $Q(x)$.

Why It Matters

These equations are the primary language for describing “accumulation” and “decay” in the real world. They model everything from the temperature of a cooling engine to the growth of a savings account. Mastering the “integrating factor” method gives you the power to project the future state of a system based on its current rate of change, providing a deterministic bridge between today’s dynamics and tomorrow’s results.

Core Concepts

• Integrating Factor ($v(x)$): A function, defined as $v(x) = e^{\int P(x) dx}$, used to transform the left side of the equation into the derivative of a product.
  • How to read: “The function v of x equals e raised to the integral of P of x with respect to x.”
  • Meaning: Multiplying by this factor turns the left side into a single derivative—setup for direct integration.
• Product Rule Reverse: Multiplying the standard form by $v(x)$ results in $\frac{d}{dx}[v(x)y] = v(x)Q(x)$.
  • How to read: “The derivative with respect to x of the quantity v of x times y equals v of x times Q of x.”
  • Meaning: After multiplying by the integrating factor, the left side is an exact derivative (product rule in reverse).
• General Solution: The solution is found by integrating both sides: $y = \frac{1}{v(x)} \int v(x)Q(x) dx$.
  • How to read: “The function y equals the ratio of one divided by v of x, multiplied by the integral of the quantity v of x times Q of x, with respect to x.”
  • Meaning: Integrate the transformed equation, then divide by $v(x)$ to solve for $y$.

Connected Concepts

• Nonhomogeneous Linear Differential Equations
• Second-Order Linear Homogeneous Equations
• Linear Equations, Problem Solving in Mathematics