# Integral curve

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The graph of a solution of a normal system

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

are the circles , where is an arbitrary constant. The integral curve is often identified with the solution. The geometric meaning of the integral curves of a scalar equation

 (*)

is the following. The equation (*) defines a direction field on the plane, that is, a field of direction vectors such that at each point the tangent of the angle of inclination of the vector with the -axis is equal to . 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 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)