Linear differential equation

A differential equation in which the unknown function and its derivatives appear linearly, with coefficients that may depend on the independent variables.

Differential equation: An equation involving an unknown function and one or more of its derivatives. Linear differential equations are a structured subclass of differential equations.
Linear differential operator: A linear map built from differentiation and coefficient functions, acting on functions. It packages the unknown function and derivatives into one linear expression.
Harmonic oscillator: A system that oscillates under a restoring force proportional to displacement. Its undamped motion satisfies a second-order homogeneous linear equation.
Nonlinear differential equation: A differential equation in which the unknown function or its derivatives enter nonlinearly. Products or nonlinear functions of the unknown distinguish it from the linear case.
Existence and uniqueness theorem: A theorem specifying conditions under which a differential equation has a solution and that solution is unique. For linear initial value problems, continuous coefficients provide standard existence and uniqueness guarantees.
Ordinary differential equation: A differential equation involving derivatives with respect to a single independent variable. This is the one-variable setting for linear differential equations.
Homogeneous differential equation: A differential equation whose forcing term is zero. Its solution space is closed under linear combinations, unlike a forced equation’s full solution set.
RC circuit: An electrical circuit containing a resistor and capacitor, with voltage evolving through charge storage and dissipation. Its voltage obeys a first-order linear equation under an applied source.
Quasilinear partial differential equation: A partial differential equation linear in its highest-order derivatives but potentially nonlinear in lower-order terms. It preserves linearity only in the highest derivatives, not across the entire equation.
Matrix exponential: A matrix function defined by the power series for the exponential, used to solve constant-coefficient linear systems. It expresses solutions of linear systems of ordinary differential equations.

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