Search results
Results from the WOW.Com Content Network
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 ...
When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution. 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]
This is a list of named linear ordinary differential equations. A–Z. Name Order Equation Applications Airy: 2 = [1] ...
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:
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, [8] meaning here that from its own dynamics, the system will reach the value zero at an ending time and stay there in zero forever after.
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 (+ + + +) =
Plus, the origin behind the phrase 'Beware the Ides of March.'
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 ...