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integrals, even if those integrals can not be expressed through
elementary functions.
As a result of finding of indefinite integrals we get
integrations primitive within arbitrary constant, it is even visible
from the simplest examples, that differential equation must have
the great number of decisions.
dy x
For example, equation  e has obviously at x
R
dx
great number of decisions, which are determined by a formula
x
y  e  , C where C - the arbitrary became.
Consequently, as a result of integration of differential
equation of the first order we will get a function, that depends
yet and from arbitrary constant C :

y   , Cx , (1.5)

that is actually we get family of decisions. Such family of
decisions is named the common decision, and if came to
expression  , x y ,C  0 use a term general integral.

1.2 Solving Cauchy Task and Its Geometrical
Interpretation

Often at solving of theoretical or practical tasks it is needed
to find not all upshots of the given equation, and only decision,
that satisfies some additional terms. One of tasks of such type is
a task - so called task Cauchy very important in the theory of
differential equations:
  y  f  , yx ,
 (1.6)
 y    yx 0 0 .

Essence of task Cauchy consists in that from all decisions
of equation (1.3) it is needed to find such decision y  y  x
which at the set independent variable value x  x gets the set
0

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