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Cauchy Theorem if a function f (x , y ,  y ,..., y (n ) 1  ) as
function of n+1 variable is continuous and has continuous
derivative parts on arguments y ,  y ,..., y (n ) 1  in some region
D  R n  1 that contains a point M (x , y ,  y ,..., y (n ) 1  ) there is
0 0 0 0 0
the unique decision of equation (3.2), that satisfies initial
conditions (3.3).
Definition 3.4 A function is named the common decision of
differential equation (3.2)

y=φ(x, c 1,c 2,…,c n), (3.4)

which depends on n arbitrary constant c 1,c 2,…,c n such, that:
1) at the arbitrary concrete set of constant c 1,c 2,…,c n she is
the decision of differential equation (3.2);
2) for any point M (x , y ,  y ,..., y (n ) 1  ) from the region
0 0 0 0 0
D, indicated in a Cauchy theorem, constant c 1,c 2,…,c n it is
possible to pick up so, that a function (3.4) will satisfy initial
conditions (3.3).
Definition 3.5 Every function which turns out from the
common decision at the concrete values of constant c 1,c 2,…,c n it
is named the partial decision of differential equation (2.2).
For differential equation of the second order of finding of
partial decision it is possible to carry out also by setting of
regional terms; the values of function in two different points
y (x )  y , y (x )  y are thus set .
1 1 2 2
With such task it is possible to clash in the course of
resistance of materials at the study of equation of bending of
beam and in a course mathematical physics. For example, the
partial decision of Equation y '' x after regional terms
y ) 1 (  , 0 ) 2 ( y  0 is as follows. For the first we will find the
common decision of the given differential equation. For this
purpose, consistently integrating, we will find at first first
x 2
derivative: ' y   C , and then and function:
1
2

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