# General integral

of a system of $n$ ordinary differential equations

$$\tag{1 } x _ {i} ^ \prime = f _ {i} ( t , x _ {1} \dots x _ {n} ) ,\ \ i = 1 \dots n ,$$

in a domain $G$

The set of $n$ relations

$$\tag{2 } \Phi _ {i} ( t _ {i} , x _ {1} \dots x _ {n} ) = C _ {i} ,\ \ i = 1 \dots n ,$$

containing $n$ parameters $( C _ {1} \dots C _ {n} ) \in C \subset \mathbf R ^ {n}$, and describing in implicit form the family of functions forming the general solution of this system in the domain $G$. Often the set of functions

$$\tag{3 } \Phi _ {i} ( t , x _ {1} \dots x _ {n} ) ,\ \ i = 1 \dots n ,$$

is called the general integral of , rather than the equations (2). Each of the equations (2) (or each function (3)) is called a first integral of . Sometimes a general integral of

means a more general set of equations than (2),

$$\Phi _ {i} ( t , x _ {1} \dots x _ {n} , C _ {1} \dots C _ {n} ) = 0 ,\ i = 1 \dots n .$$

For an $n$- th order ordinary differential equation

$$y ^ {(} n) = f ( x , y , y ^ \prime \dots y ^ {(} n- 1) )$$

a general integral is a single relation with $n$ parameters,

$$\Phi ( x , y , C _ {1} \dots C _ {n} ) = 0 ,$$

describing the general solution of this equation in the domain $G$ in the form of an implicit function.

A general integral of a first-order partial differential equation is a relation between the variables in the equation involving one arbitrary function such that the equation is satisfied when the relation is substituted in it, for every choice of the arbitrary function.