Differential equation

A differential equation relates an unknown function to one or more of its derivatives. Its solutions are functions that satisfy this relation, often describing how quantities change.

Derivative: A measure of how a function changes with respect to one of its variables. Differential equations use derivatives to encode rates of change.
Separation of variables: A method for solving certain differential equations by placing each variable and its differential on opposite sides. It solves equations whose variables can be separated into independent factors.
Newton's second law: The law stating that an object's net force equals its mass times its acceleration. Because acceleration is a second derivative, the law produces equations of motion.
Algebraic equation: An equation relating variables through algebraic operations rather than derivatives or integrals. Unlike a differential equation, it constrains values without specifying rates of change.
Ordinary differential equation: A differential equation involving an unknown function of one independent variable and its ordinary derivatives. It is the single-variable case of the general concept.
Integrating factor: A function that multiplies a differential equation to make it easier to solve, often by turning one side into a product derivative. It converts a first-order linear equation into an integrable form.
Heat equation: A partial differential equation describing how temperature diffuses through a material over time. It is a canonical model of change distributed through space.
Integral equation: An equation in which an unknown function appears inside an integral. It encodes relationships through accumulated values rather than derivatives.
Partial differential equation: A differential equation involving an unknown function of multiple variables and its partial derivatives. It extends the concept to functions varying across several independent variables.
Existence and uniqueness theorem: A theorem establishing conditions under which a differential equation has a solution and that solution is unique. It tells when initial data determine a well-defined trajectory.

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