# 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)

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
How to Cite This Entry:
Integral curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_curve&oldid=47368
This article was adapted from an original article by N.N. Ladis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article