linear differential equation: meaning, definition, pronunciation and examples

A differential equation is linear if the unknown function and all its derivatives appear only to the first power (are not squared, cubed, etc.) and are not multiplied by each other. The equation can be written as a sum of terms, each being a coefficient (which may depend on the independent variable) times the function or one of its derivatives.

Linear differential equations obey the principle of superposition (solutions can be added together to form new solutions) and generally have well-established, systematic solution methods. Nonlinear differential equations, where the unknown function or its derivatives are raised to a power other than one or multiplied together, are often more complex to solve and may exhibit chaotic behaviour.

They are used extensively in modelling mechanical vibrations (spring-mass systems), electrical circuits (RLC circuits), heat conduction, wave propagation, population dynamics under ideal conditions, and in control systems theory.

Yes, absolutely. The 'linear' refers to the form of the equation itself, not the shape of its solution graph. Solutions to linear differential equations are often curves, such as exponentials, sine waves, or polynomials.

A type of mathematical equation that relates a function to its derivatives, where both the function and its derivatives appear only to the first power and are not multiplied together.

Linear differential equation is usually technical/formal in register.

Linear differential equation: in British English it is pronounced /ˌlɪn.i.ə ˌdɪf.əˌren.ʃəl ɪˈkweɪ.ʒən/, and in American English it is pronounced /ˌlɪn.i.ɚ ˌdɪf.əˌren.ʃəl ɪˈkweɪ.ʒən/. Tap the audio buttons above to hear it.