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we come to equation of the n-k -th order of relatively unknown
function of z(x):

( zxF , , ,  z ,   z ..., z ( kn ) )  , 0 (3.10)

that is we bring an order down of equation on k units. If it is
succeeded to find the common decision of this equation in a
kind z   (x ,c ,c ,...,c ), we will get differential equation of
1 2 n k
previous kind
z  y (k )   (x ,c ,c ,...,c ), (3.11)
1 2 n k

the decision of which is found k-times integration. In particular,
if n=2, k=1 equation (3.11) – first order.
On occasion finding the decision as an explicit or non-
obvious function is difficult, but it is succeeded to find decision
in a parametric form.
We will consider examples, which illustrate the considered
types of equations of higher order and methods of their solving.

Example 3.3 To find the common decision of equation
y    ctg x    y  . 2

 The given equation is equation of the considered type,
where n=3, k=2. We enter a new unknown function z  y  and
get linear equation of the first order ctgz x  z  , 2 which write
down in a kind z   z tg x  2tg . x

Its common decision:


 tgxdx  tgxdx ln cos x  ln cos x
z  e (  2tgx e dx  c 1 )  e 2 (  tgx e dx  c 1 ) 
tgx sin x dx
 cos x 2(  dx  c )  cos2 x   c cos x 
1
1
cos x cos 2 x

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