First Integral

First Integral

 

The first integral of a system of ordinary differential equations

is a relation of the form

ϕ (x, y1, …, yn) = C

where C is an arbitrary constant. The left side of the relation remains constant when any solution y1 = y1(x), …, yn = yn(x) of the system is substituted but is not a fixed constant. Geometrically, the first integral is a family of hypersurfaces in the (n + 1)-dimensional space Oxy1yn, in each of which there exists a subfamily of the integral curves of the system. For example, y2 + z2 = C2 (circular cylinder) is a first integral of the system dy/dx = z, dz/dx = – y; the integral curves y = C sin (xx0) and z = C cos (xx0) are helices on the cylinders (see Figure 1). If k independent first integrals ϕi (x, y1 …, yn) = Ci (i = 1, …, k; k < n) of a system are

Figure 1

known, the system’s order can in general be decreased by k units. If k = n, the general integral of the system is obtained without integration.

REFERENCE

Stepanov, V. V. Kurs differentsial’nykh uravnenii, 8th ed. Moscow, 1959.
References in periodicals archive ?
While Flobert's invention would lead to the rimfire cartridges still in wide use today, it wasn't the first integral metallic cartridge in the history of firearms.
Then, up to a bimeromorphic transformation, X is holomorphic, X has a first integral or S has a rational or elliptic fibration preserved by X (with each component of the locus of poles of X contained in a fiber).
Using the Lagrangian of plane and spherical symmetric static spacetimes, a complete list of Noether symmetries have been obtained and new first integral have been found.
We are delighted to have Union Insurance as our first Integral client in the Middle East and to be their partner for business transformation.
A function [PHI] : U [right arrow] C (U an open set) is said to be a first integral of (1), if [PHI](z) is a constant along any solution curve of system (1).
BTCS' investment in Spondoolies comes on the heels of an April 28 announcement of the intention for BTCS and Spondoolies to merge and is serving as the first integral step of the planned merger.
Now we take the first integral of (5) as being the process of creation of a virtual particle-antiparticle pair.
It is proved that nonlinear mechanical systems with an analytical first integral allow periodic solutions which tend towards linear normal vibration modes as amplitudes tend to zero.
Either one can be solved by hand; however, doing so requires time and patience: the first integral requires applying integration by parts twice, while the second integral involves a process of long division.
During the last few years, the numerical methods and exact solution methods have been proposed to solve fractional differential equations, for example, the Adomian decomposition method [10], the homotopy perturbation method [11, 12], the variational iteration method [13, 14], the differential transform method [15,16], the G'/G method [17, 18], the first integral method [19], and the exp-function method [20].
3] [right arrow] R non-constant in any open subset of U is a first integral if H (x(t), y(t), z(t)) is constant on all solutions (x(t), y(t),z(t)) of [chi] contained in U, i.
It is easy to show that the first integral can be computed as

Full browser ?