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general integral[¦gen·rəl ′int·ə·grəl]
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.