Differentialekvationssystem

Differentialekvationssystem, or a system of differential equations, refers to a collection of two or more interrelated differential equations that describe how several unknown functions evolve over an independent variable, typically time. The equations are linked because each derivative may depend on the unknowns in the system. A common form is dx/dt = F(x,t), where x(t) is a vector of n unknown functions and F maps R^n × R to R^n.

Systems can be autonomous (F depends only on x) or non-autonomous (depends on t). They can be linear (dx/dt = A x + b(t)) or nonlinear. Any higher-order system can be rewritten as a first-order system by introducing new variables for derivatives.

Solution approaches include analytical methods and numerical techniques. Analytical methods cover solving linear systems by finding eigenvalues and eigenvectors of the matrix A; in nonlinear cases closed-form solutions may not exist. Qualitative analysis studies equilibria x* with F(x*) = 0, stability, and phase portraits. Existence and uniqueness theorems (such as Picard–Lindelöf) guarantee a local solution under certain conditions. Numerically, methods like Euler or Runge-Kutta approximate trajectories; stiff systems may require implicit schemes.

Initial value problems specify x(t0) = x0, while boundary value problems impose conditions at multiple points, common in systems described by partial differential equations (PDEs) rather than ordinary differential equations (ODEs). In PDE contexts, boundary or initial data determine the solution over a domain.

Applications of differentialekvationssystem span physics, engineering, biology, economics, and beyond. They model coupled oscillations, chemical reaction networks, population dynamics, electrical circuits, control systems, and many processes where multiple quantities influence each other over time.