Integral of a differential equation

A solution of the differential equation. By an integral of a differential equation is primarily meant a relation of the form $ \Phi ( x , y ) = 0 $ defining a solution $ y $ of an ordinary differential equation

$$ \tag{1 } F ( x , y , y ^ \prime \dots y ^ {(} n) ) = 0 $$

as an implicit function of the independent variable $ x $. The solution is in this case also said to be a particular integral, in contrast to the general integral of equation (1), that is, a relation

$$ \tag{2 } \Phi ( x , y , C _ {1} \dots C _ {n} ) = 0 , $$

from which one can obtain by an appropriate choice of the constants $ C _ {1} \dots C _ {n} $ any integral curve of (1) lying in some given region $ G $ of the $ ( x , y ) $- plane. If the arbitrary constants $ C _ {1} \dots C _ {n} $ are eliminated from equation (2) and the $ n $ relations obtained from it by repeated differentiation with respect to $ x $( where $ y $ is regarded as a function of $ x $), then equation (1) results. A relation of the form

$$ \tag{3 } \Phi ( x , y , y ^ \prime \dots y ^ {(} k) ,\ C _ {1} \dots C _ {n-} k ) = 0 , $$

containing derivatives up to order $ k $, $ 1 \leq k < n $, and $ n - k $ arbitrary constants, arising from the process of integrating equation (1), is sometimes called an intermediate integral of equation (1). If an intermediate integral (3) is known, then the solution of equation (1) of order $ n $ is reduced to the solution of equation (3) of order $ k $. If (3) contains just one arbitrary constant, that is, if $ k = n - 1 $, then it is called a first integral of (1). This equation has exactly $ n $ independent first integrals; knowledge of such integrals enables one to obtain the general solution of (1) by eliminating the quantities $ y ^ \prime \dots y ^ {(} n- 1) $ from them.

If one considers a first-order system of ordinary differential equations,

$$ \tag{4 } \frac{d x _ {i} }{dt} = \ f _ {i} ( t , x _ {1} \dots x _ {n} ) ,\ \ i = 1 \dots n , $$

then by a general integral of it is meant a set of relations

$$ \tag{5 } \Phi _ {i} ( t , x _ {1} \dots x _ {n} ) = C _ {i} ,\ \ i = 1 \dots n , $$

where the $ C _ {i} $ are arbitrary constants, which describes in implicit form all the solutions of the system (4) in some region $ G $ of the $ ( t , x _ {1} \dots x _ {n} ) $- space. Each of the relations (5) is itself called a first integral of the system (4). More often, by a first integral of the system (4) one means a function $ u ( t , x _ {1} \dots x _ {n} ) $ with the property that it is constant along any solution of the system (4) in a region $ G $. The system (4) has exactly $ n $ independent first integrals, knowledge of which enables one to find the general solution without integrating the system; knowledge of $ k $ independent first integrals enables one to reduce the solution of the system (4) of order $ n $ to the solution of a system of order $ n - k $. A smooth function $ u ( t , x _ {1} \dots x _ {n} ) $ is a first integral of the system (4) with smooth right-hand side if and only if it satisfies the equation

$$ \frac{\partial u }{\partial t } + \sum _ { i= } 1 ^ { n } f _ {i} ( t , x _ {1} \dots x _ {n} ) \frac{\partial u }{\partial x _ {i} } = 0 . $$

Similar terminology is sometimes used in the theory of first-order partial differential equations. Thus, by an integral of the differential equation

$$ \tag{6 } F \left ( x , y , z ,\ \frac{\partial z }{\partial x } ,\ \frac{\partial z }{\partial y } \right ) = 0 , $$

or by a particular integral of it, is meant a solution of this equation (an integral surface). By a complete integral of (6) is meant a family of solutions $ \Phi ( x , y , z , a , b ) = 0 $ depending on two arbitrary constants. A general integral of equation (6) is a relation containing one arbitrary function and giving a solution of the equation for each choice of this function.

References

[1]	W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)

References

[a1]	K. Rektorys (ed.) , Survey of applicable mathematics , Iliffe (1969) pp. Sects. 17.2, 17.8, 17.18, 17.20
[a2]	E.L. Ince, "Integration of ordinary differential equations" , Oliver & Boyd (1956)