Ordinary differential equation

An ordinary differential equation relates an unknown function of one independent variable to that function’s derivatives. Its solutions are functions that satisfy the relation on an interval.

Derivative: A measure of how a function changes with respect to its input. Derivatives of the unknown function are the quantities an ODE relates.
Separable differential equation: A first-order differential equation whose variables can be separated into factors of the independent and dependent variables. Separation solves a common ODE form by integrating each variable’s side.
Newton's laws of motion: Classical laws relating forces to the motion of objects. Newton’s second law often becomes an ODE for position or velocity over time.
Partial differential equation: An equation involving an unknown function of multiple independent variables and its partial derivatives. A PDE has multiple independent variables, unlike an ODE’s single independent variable.
Order of a differential equation: The highest derivative appearing in a differential equation. The highest derivative sets an ODE’s order and often determines how many initial values are needed.
Linear differential equation: A differential equation in which the unknown function and its derivatives appear linearly. Linearity enables superposition and systematic solution methods.
Harmonic oscillator: A system whose restoring force is proportional to displacement from equilibrium. Its motion is governed by a second-order linear ODE.
Difference equation: An equation relating values of an unknown sequence at different discrete indices. Difference equations describe discrete updates rather than derivatives with respect to a continuous variable.
General solution: A family of functions that satisfies a differential equation and includes arbitrary constants or functions. Solving an ODE often means finding a family that covers its possible solutions.
Existence and uniqueness theorem: A theorem giving conditions under which a differential equation has a solution and that solution is unique. It determines whether initial data actually specify a well-defined solution.

Linked from 62 pages