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Mingarelli Identity
In the field of ordinary differential equations, the Mingarelli identityThe locution was coined by Philip Hartman, according to is a theorem that provides criteria for the oscillation theory, oscillation and oscillation theory, non-oscillation of solutions of some linear differential equations in the real domain. It extends the Picone identity from two to three or more differential equations of the second order. The identity Consider the solutions of the following (uncoupled) system of second order linear differential equations over the –interval : :(p_i(t) x_i^\prime)^\prime + q_i(t) x_i = 0, \,\,\,\,\,\,\,\,\,\, x_i(a)=1,\,\, x_i^\prime(a)=R_i where i=1,2, \ldots, n. Let \Delta denote the forward difference operator, i.e. :\Delta x_i = x_-x_i. The second order difference operator is found by iterating the first order operator as in :\Delta^2 (x_i) = \Delta(\Delta x_i) = x_-2x_+x_,, with a similar definition for the higher iterates. Leaving out the independent variable f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematics), function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equation, ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with stochastic differential equations, ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations 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 :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |