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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] ...
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 .
1.6 Ordinary Differential Equations (ODEs) 1.7 Riemannian geometry. 2 Physics. Toggle Physics subsection. ... Linear-quadratic regulator; Matrix differential equation;
Most ODEs that are encountered in physics are linear. Therefore, most special functions may be defined as solutions of linear differential equations (see Holonomic function ). As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression , numerical methods are commonly used for solving differential ...
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 ...
Nonlinear ones are of particular interest for their commonality in describing real-world systems and how much more difficult they are to solve compared to linear differential equations. This list presents nonlinear ordinary differential equations that have been named, sorted by area of interest.