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These solutions are called the trivial solutions. For example, consider the differential equation ′ = where = is a function whose derivative is ′. The trivial solution is the zero function = while a nontrivial solution is the exponential function
The differential equation is said to be in Sturm–Liouville form or self-adjoint form.All second-order linear homogenous ordinary differential equations can be recast in the form on the left-hand side of by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of second-order partial differential equations, or if y is a vector).
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown (s) consists of one (or more) function (s) and involves the derivatives of those functions. [ 1 ] The term "ordinary" is used in contrast with partial differential equations (PDEs ...
Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form. with and being a piecewise continuous periodic function with period and defines the state of the stability of solutions. The main theorem of Floquet theory, Floquet's theorem, due ...
Differential-algebraic system of equations. In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. The set of the solutions of such a system is a differential algebraic variety, and corresponds to an ideal ...
Given a path F(t), the ordinary differential equation = /, with initial condition () =, has a unique C 1 solution g(t) with values in G, giving the lift by parallel transport of F. If Q(t) is the corresponding path of rank 2 projections, the conditions for parallel transport are
A Green's function, G(x,s), of a linear differential operator L = L(x) acting on distributions over a subset of the Euclidean space , at a point s, is any solution of. (1) where δ is the Dirac delta function. This property of a Green's function can be exploited to solve differential equations of the form. (2) If the kernel of L is non-trivial ...
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.
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