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Lecture 2 Solving of Some Equations of the First Order,
That Get Untied in Relation to Derivative

2.1 Equation in Complete Differentials Equation

Definition 2.1  , yxP dx  Q  , yx dy  0 (2.1)

it is named equation in complete differentials, if him left
chastina is the complete differential of some function  yxu , ,
that is, if
u  u 
.  yxP , dx  Q  yx, dy  dx  dy  du (2.2)
x  y 

It is possible to show that implementation of equality is the
necessary and sufficient condition of existence of such function
u  yx, 
 P  Q
.  (2.3)
y  x 

If a condition (2.3) takes place, equation (2.1) acquires a
kind du  0 , and his general integral will be  yxu ,  C .
Consequently, a basic task consists in finding of function
u  yx, . We will do it in such sequence. With (2.2) we write
down to equality
u
  P  , yx ,
x
 (2.4)
 u  Q  , yx .
y


Deciding the first from these equations by integration on a
variable x , we will get

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