General Integral


general integral

[¦gen·rəl ′int·ə·grəl] (mathematics) general solution

General Integral

 

A general integral of an ordinary differential equation

F(x, y, y’, …,y(n)) = 0

is a relation

Φ(x, y, C1, …,Cn) = 0

containing n essential, arbitrary constants C1, …, Cn, which implies the given differential equation (seeDIFFERENTIAL EQUATIONS). In other words, the differential equation is the result of the elimination of the constants Ci (i = 1, …, n) from the system of equations

These constants are essential in the sense that the process of eliminating them from the system (*) does not lead to a differential equation different from the given one. There is a close connection between a general integral and a general solution. If the constants Ci belonging to a general integral are assigned specific values, then a particular integral is obtained. The incomplete elimination of the constants Ci from the system (*) yields an intermediate integral

F(x, y, y’, …,y(n–k), C1, …,Cn) = 0

where 1 ≤ k ≤ n – 1; in particular, for k = 1 we obtain a first integral. Geometrically, a general integral is an n-parameter family of integral curves.

REFERENCE

Stepanov, V. V. Kurs differentsial’nykh uravnenii, 8th ed. Moscow, 1959.