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that converts (7.3) into an identity at any values arbitrary
constant C . Thus, what no initial conditions were:
i

y ( )x  y y ( )x  y  , y ( )x  y (7.5)
1 0 10 , 2 0 20 , n 0 n 0 ,

the these constant C are determined simply. The adopted
i
system of functions (7.4) at down a condition (7.5) is laid is the
common decision. Any n the functions, which turn out from
the common decision (6.4) at the concrete values constant
C ,C , ,C are named the partial decision of the system of
1 2 n
differential equations.
The search of partial decision of the system of equations
(7.3) at initial conditions (7.5) is named Cauchy task. We will
formulate a theorem about existence and unique of decision
Cauchy tasks are of the system of differential equations.
Cauchy Theorem The decision of task Cauchy exists and
)
unique, if functions f ( ,x y , , y ), , f ( ,x y , , y
1 1 n n 1 n
continuous together with derivative parts on arguments
y 1 , , y in the region of D (n  1)  measurable space, which
n
contains a point M ( ,x y , , y ).
0 0 10 n 0

7.2 Solving of the Normal System of Differential
Equations

We already noticed about possibility of "development" of
differential equation of the n-th order in the system of
differential equations of the first order. In the turn at certain
terms the normal system of equations (7.3) can be taken to
equivalent one differential equation of th order with one sought
function after. It will be one of methods of solving (integration)
of such system of equations so called method of exceptions.
It is carried out thus. We differentiate the first equation of
the system (7.3) on x (we find from both parts of the first
equality complete derivative whit respect of x ):

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