Page 8 - 4549

Lecture 1 Notion of Ordinary Differential Equations

At studying of many phenomena of nature, at research of
evolutional processes there are many technical and physical
tasks at solving of which not always is succeeded to set direct
dependence between sizes which describe that or other process.
However in most cases it is succeeded to set relation between
sizes (by functions) and speeds of their change in relation to
other independent variables of sizes. That is to get equation, in
what unknown functions enter under the sign of derivative. Such
equations are named differential.

1.1 Common Notions and Determinations

Definition 1.1 It is named ordinary differential equation of
the n -th order of relatively unknown function   xy to
expression
 , yxF ,  y ,  y ,..., y  n  0 (1.1)
between an independent variable x , by a function   xy and its
derivatives , yy ,   ...y  n .
Definition 1.2 The greatest order of derivative unknown
function is named the order of differential equation that is
contained in correlation (1.1).
We will consider differential equations of the first order, the
general type of which can be written down in a kind

 , yxF ,  y  0 (1.2)

that expresses communication between an independent
variable x , by the sought function   xy and its derivative .y
Equation (1.2) can not contain in an obvious kind x or y ,
but necessarily it is to contain derivative y (differently
expression stops to be differential equation). If equation (1.2) is
succeeded to untie in relation to derivative, we will get more
comfortable kind

y  f  yx,  (1.3)
6