Ordinary Differential Equations glossary

Clear, one-line definitions of the Ordinary Differential Equations terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

differential equation

An equation relating an unknown function to its derivatives; ordinary if in one variable, partial if in several.

See: What Is a Differential Equation?, Methods for Ordinary Differential Equations

homogeneous

A linear differential equation is homogeneous when its right-hand side is zero; solutions then form a vector space.

initial value problem

An ODE together with prescribed values of the solution (and its derivatives) at a single point; admits a unique solution under mild hypotheses.

integrating factor

A function μ(x) that multiplies a linear first-order ODE y' + p(x)y = q(x) to make the left side an exact derivative; μ = exp(∫p dx).

linear

A differential equation or operator is linear when it depends linearly on the unknown function and its derivatives, obeying superposition.

See: Linear algebra primer, Hudson and Linear Slip

linear differential equation

A differential equation of the form a_n(x)y^(n) + ... + a_1(x)y' + a_0(x)y = f(x); linear in y and its derivatives.

nonlinear differential equation

A differential equation that is not linear in the unknown function or its derivatives; superposition of solutions generally fails.

order

The order of a differential equation is the highest derivative of the unknown that appears in it.

ordinary differential equation

A differential equation for a function of a single independent variable, involving only ordinary (not partial) derivatives.

See: Methods for Ordinary Differential Equations

separable equation

A first-order ODE that can be written dy/dx = g(x)h(y), so the variables separate to h(y) dy = g(x) dx and each side integrates independently.