Differentialligning

Differentialligning, in mathematics, is a relation that involves an unknown function and its derivatives. Such equations describe how a quantity changes in response to its own rate of change. An ordinary differentialligning (ODE) depends on a single independent variable, while a partial differentialligning (PDE) involves several independent variables and partial derivatives. The term differentialligning is commonly used in Danish-language texts for what English speakers call a differential equation.

Equations may be first order or higher order. A first-order separable ODE has the form dy/dx = f(x, y) and can often be solved by separation of variables. Linear first-order: y' + p(x) y = q(x). Higher-order linear equations have constant or variable coefficients, such as a_n y^(n) + ... + a_1 y' + a_0 y = g(x). Partial differential equations involve partial derivatives, with classical examples including Laplace's equation ∇^2 φ = 0, the heat equation ∂t u = κ ∇^2 u, and the wave equation ∂tt u = c^2 ∇^2 u.

Solutions may be obtained analytically, by methods such as integrating factors, characteristic equations, or separation of variables, when possible. For many equations, especially nonlinear or complex PDEs, numerical methods are used, including Euler's method and higher-order Runge–Kutta schemes. The existence and uniqueness theorem (Picard–Lindelöf) provides conditions under which a solution to an initial value problem exists and is unique.

Differentialligninger have wide-ranging applications in physics, engineering, biology, and economics, modeling motion, population dynamics, heat and mass transfer, and financial systems. The subject dates to the work of Newton and Leibniz and was developed further in the 18th and 19th centuries. Modern theory covers qualitative behavior, stability, and numerical approximation, reflecting the central role of differentialligning in mathematical modeling.