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Lecture 3 Differential Equations of Higher Orders

3.1 Common Notions and Determinations

Definition 3.1 Equation
(xF , y ,  y ,..., y (n ) )  0 , (3.1)

where x is an independent variable, y – the sought function
after, and function F is certain and continuous in some region

GR n+2 (n 1 ) and necessarily depends on y (n ) is named
ordinary differential equation of the n  thorder.

Definition 3.2 The differential equation of the n-th order,
untied relatively senior derivative, has a kind
y (n )  f (x , y ,  y ,..., y (n ) 1  ), (3.2)
where f is the function of f is also considered continuous in
some region D  R n  1 of change of the arguments.

Definition 3.3 A function у(х) is named the decision of
equation (3.2) on an interval (a, b) in(х), which satisfies terms:
1) у(х) n times continuously differentiated on (a, b);
2) ( yx , (x ), y (x ),..., y (n ) 1 (x )) D x  (a ,b );
3) y(x) converts equation (2) into an identity, that is
y (n ) (x )  f (x , y (x ), y (x ),..., y (n ) 1 (x )) x  (a ,b ) .
The decision of equation is like determined (3.1).
By a task Cauchy for equation (3.1) or (3.2) the task of
finding of decision is named у(х) equation (3.1) or (3.2), that
satisfies initial conditions:
(xy )  y , y (  x )  ,  y ..., y (n ) 1  (x )  y (n ) 1  , (3.3)
0 0 0 0 0 0
where x  (a ,b ), y ,  y ,..., y (n ) 1  – the set numbers.
0 0 0 0
The theorem of existence and unique of decision of task
Cauchy (3.2) takes place – (3.3).

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