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In obedience to our agreement, we will stop on a case
n  2 . Lets y 1 ( ) ,x y 2 ( )x - linearly independent upshots of
equation (4.4), then after property of decisions the LHDE
function
y  C y ( )x  C y ( )x (4.13)
1 1 2 2

also will be the decision of this equation. Consequently, it is
enough to show for proof of theorem, that for arbitrary initial

conditions ( )y x  y , ( )y x  y , it is possible to pick up to
0 0 0 1
,
steel C C
1 2
so that a function (4.13) was the decision of Cauchy task.
We have, thus, for determination C ,C the system of
1 2
equations
 C y ( )x  C y 2 ( )x 0  y 0 ,
0
2
1 1
 (4.14)
 C y ( )x  C y 2 ( )x 0  y 1 .
2
0
1

A Wronski-Determinante W ( )x is the determinant of this
0
system . Such determinant, in obedience to a theorem 3, does
not equal a zero. Consequently, the system of upshots has.
Note 4.1 There is no regular reception for finding of the
fundamental system of decisions LHDE, and therefore and
finding of common decision of such Equations. Must be limited
to the searches, which are based on conjecture shrewdness.
Moreover, a lot of linear homogeneous differential equations
(with variable coefficients) in general are not integrated in
elementary functions. The exception is made only by
homogeneous equations with constant coefficients. Before
integration of such equations we will pass on a next lecture.
There is useful a next theorem.
Theorem 4.5 If one decision y  part LHDE-2 is
0
1
known –
y  p y  p y  0 ,
1 2
Finding of its general decision with integration of functions.

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