Search results
Results from the WOW.Com Content Network
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. [ 1 ]
William Wordsworth's Ode on Intimations of Immortality from Recollections of Early Childhood (1807) and Thomas Gray's The Progress of Poesy: A Pindaric Ode (1757) are both written in the Pindaric style. Gray's The Bard: A Pindaric Ode (1757) is a Pindaric ode where the three-part structure is thrice repeated, yielding a longer poem of nine stanzas.
An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Thus x is often called the independent variable of the equation.
The step size is =. The same illustration for = The midpoint method converges faster than the Euler method, as .. Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs).
Differential equations are prominent in many scientific areas. 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.
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:
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
The ode generally has three parts: a strophe, an antistrophe, and an epode. The strophe and the antistrophe of the ode possess similar metrical structures and, depending on the tradition, similar rhyme structures. In contrast, the epode is written with a different scheme and structure.