Integral curve

The graph of a solution $ y = y ( x) $ of a normal system

$$ y ^ \prime = f ( x , y ) ,\ y \in \mathbf R ^ {n} , $$

of ordinary differential equations. For example, the integral curves of the equation

$$ y ^ \prime = - \frac{x}{y} $$

are the circles $ x ^ {2} + y ^ {2} = c ^ {2} $, where $ c $ is an arbitrary constant. The integral curve is often identified with the solution. The geometric meaning of the integral curves of a scalar equation

$$ \tag{* } y ^ \prime = f ( x , y ) $$

is the following. The equation (*) defines a direction field on the plane, that is, a field of direction vectors such that at each point $ ( x , y ) $ the tangent of the angle of inclination of the vector with the $ x $- axis is equal to $ f( x , y ) $. The integral curves of (*) are then the curves that at each point have a tangent coinciding with the vector of the direction field at this point. The integral curves of (*) fill out the entire region in which the function $ f ( x , y ) $ satisfies conditions ensuring the existence and uniqueness of the Cauchy problem; the curves nowhere intersect one another and are nowhere tangent to one another.

References

[1]	I.G. Petrovskii, "Ordinary differential equations" , Prentice-Hall (1966) (Translated from Russian)

Comments

A normal system of differential equations is a system of differential equations of the form

$$ \frac{d ^ {n _ {k} } x _ {k} }{d t ^ {n _ {k} } } = $$

$$ = \ F _ {k} \left ( x _ {1} , \frac{d x _ {1} }{d t } ,\dots; x _ {2} , \frac{d x _ {2} }{d t } ,\dots; \dots ; x _ {m} , \frac{d x _ {m} }{d t } ,\dots \right ) , $$

$ k = 1 \dots m $, such that the function $ F _ {k} $ only depends on the $ d ^ {j} x _ {i} / d t ^ {j} $ for $ j < n _ {i} $, $ i= 1 \dots m $.

References

[a1]	G. Birkhoff, G.-C. Rota, "Ordinary differential equations" , Ginn (1962) pp. Chapt. V §5
[a2]	E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) pp. §§3.6, 3.51, 4.7, A.5