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functions (xu ),v (x ) . For this purpose we will put (5.12) in (5.6)
and get
(u (x )  iv (x ) )   а (u (x )  iv (x ) )   а (u (x )  iv (x ))  0
1 2

or

(u (   x )  uа (  x )  а u (x ))  і (v (   x )  vа (  x )  а v (x ))  0
1 2 1 2

If complex function is zero, it is real and imaginary parts
also equal a zero. That is

(u (   x )  uа (  x )  а u (x ))  0 and (  xv )  vа (  x )  а v (x )  0 .
1 2 1 2

The same it is led to, that functions are decisions. (5.6).
Pursuant to the formula of Euler of decisions (5.11) it is
possible to write down in a kind:

y  e  x (cos x  i sin x ); y  e  x (cos x  i sin ) x  (5.13)
1 2

On the basis of set higher, Equations decisions (5.6) will be
real parts
x

x

y  e cos x  y ;  e sin x  (5.14)
1 2

y e  x sin x
How 2   tg x  const , functions y 1 , y are
2
y e  x cos x
1
linearly independent and form the fundamental system of
decisions. Consequently, general decision in this case will be
written down:
y  C y  C y  e  x (C cos x  C sin  ) x . (5.15)
1 1 2 2 1 2

Example 5.3 To find the common decision of equation
   y 6   y 13 y 0.

 We make characteristic equation

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