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In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form + ′ + ″ + () = where a 0 (x), ..., a n (x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y (n) are the successive derivatives of an unknown function y of ...
This is a list of named linear ordinary differential equations. A–Z. Name Order Equation Applications Airy: 2 = [1] ...
1.6 Ordinary Differential Equations (ODEs) 1.7 Riemannian geometry. 2 Physics. Toggle Physics subsection. ... Linear-quadratic regulator; Matrix differential equation;
The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE (see, for example Riccati equation). [5] Some ODEs can be solved explicitly in terms of known functions and integrals.
For an arbitrary system of ODEs, a set of solutions (), …, are said to be linearly-independent if: + … + = is satisfied only for = … = =.A second-order differential equation ¨ = (,, ˙) may be converted into a system of first order linear differential equations by defining = ˙, which gives us the first-order system:
Explicit examples from the linear multistep family include the Adams–Bashforth methods, and any Runge–Kutta method with a lower diagonal Butcher tableau is explicit. A loose rule of thumb dictates that stiff differential equations require the use of implicit schemes, whereas non-stiff problems can be solved more efficiently with explicit ...
Similar to ODEs, many properties of linear DDEs can be characterized and analyzed using the characteristic equation. [5] The characteristic equation associated with the linear DDE with discrete delays = + + + is the exponential polynomial given by (+ + + +) =
Method of lines - the example, which shows the origin of the name of method. The method of lines (MOL, NMOL, NUMOL [1] [2] [3]) is a technique for solving partial differential equations (PDEs) in which all but one dimension is discretized.