# First Integral

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## 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 ?
The MIB is the first integral brake system for motorcycles from Continental with ABS function.
In 2001, however, by invoking the 1758 Spanish decree, and because the city was the capital, Vigan was once again elevated as a ciudad becoming the first integral city of the province of Ilocos Sur.
UNPay is the world's first integral one-stop FinTech service solution and one of China and Asia's leading technology companies.
The first n quadrature weights for the first integral (5.1) are
The first integral, shown in (8) or (9), maybe quite simple for practical problems if f(x) is an analytically integrable function.
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).
It is important to mention here that under a specific parameter restriction [sigma] = [beta]([rho] + [pi])/[2[pi][beta] - [delta] + [delta][beta] - [pi]] (where [sigma] is the inverse of the elasticity of intertemporal substitution) a third first integral was obtained and the closed form solution for this case was new in the literature (see Naz et al [9]).
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.
At present, several methods for finding the exact solutions of fPDEs have been presented, for example, the Adomian decomposition method [3-6], variational iteration method [7-9], homotopy perturbation method [10-13], homotopy analysis method [14, 15], differential transform method [16], spectral methods [17, 18], discontinuous Galerkin method [19], Kansa method [20], fractional subequation method [21], generalized fractional subequation method [22], fractional projective Riccati expansion method [23], exp-function method [24, 25], (G'/G)-expansion method [26-28], functional variable method [29, 30], and first integral method [31, 32].
Divergence symmetries are thus a more relaxed form of variational symmetries, and they also give rise to conservation laws and first integral as variational symmetries do by Nrether's theorem.

Site: Follow: Share:
Open / Close