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As z  , y the last equation is Equation of the first order of
relatively unknown function у(х) which gets untied by
integration:
1 c 1 1 c 1 c 1
1
c
c
1


 y  xe 1 x , y  xe 1 x dx   xd (e 1 x )  (xe 1 x  e 1 x dx ) 
c 1 c 1
1 c 1 1 c 1 c x  1 c 1
 (xe 1 x  e 1 x ) c  2 1 e 1 x  c 2 .
c c c 2
1 1 1

Consequently, the common decision was got
c x  1
y  1 e c 1 x 1  c 2 .
c 2
1

We find the value of constant с 1 and с 2 from initial
conditions 1(y )  e , y ) 1 (  e 2 . Obsessed system of equations:
c  1 1  2 1 
e  1 e 1 c  c 2 , e  e 1 c ,
c 1 2

from which we find easily, that с 1=1, с 2=е.
Therefore the sought decision after part of the given
equation is determined by a formula
y  (x  ) 1 e x 1  . e 

Example 3.5 A body by mass m falls free from some
height without initial velocity. At falling a body tests
proportional to the square of speed of body. To find the law of
motion of body.

 Lets s  s ) (t - the way passed by a body in times of t
ds d 2 s
from the beginning of motion, then  v,  w – speed and
dt dt 2
acceleration of motion accordingly.

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