Page 11 - 4549

value y  y , where x and y - the set numbers which are
0 0 0
y
x
named the initial values, and a condition    y is named an
0 0
initial condition.
For example we shall find that decision of equation
dy  
 cos x , which satisfies an initial condition  y   . 2
dx  2 
All upshots of the given equation are determined by a
formula y  sin x  C which gives the common decision.

Decision, that answers the set initial values x  and y  2
0 0
2
will be found at some defined value C . And for this purpose it
is needed in the written down family of decisions to put initial

data: 2  sin  C , from where and we find C  . 1 At last
2
decision will be y  sin x . 1 So called partial decision of
differential equation.
Terms of existence and unique of partial decision of

differential equation y  f  yx, , that satisfies an initial
y
x
condition    y , formulated in a theorem Cauchy.
0 0
Cauchy Theorem If function  yxf ,  and its partial
derivative f   yx,  continuous in the domain region D that
у
contains a point M 0  , yx 0 0 , there is the unique decision

y  x of differential equation y  f  yx, , which satisfies an
x
initial condition    y .
y
0 0
Thus, as was earlier specified, distinguish: common
decision y   , Cx , that converts equations (1.3) in an identity
at any value of arbitrary constant C , among which the value C
0
that satisfies an initial condition    y , and decision part
y
x
0
0
y    , Cx 0 ,, is which we get from general at C  C 0 .
0
9