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0
. y  4y  3y  .

 We make characteristic equation
 2  4  3  0
Its roots real and different
  ; 1   3.
1 2
So, partial decision will be written down:
x
3
y  e 1  x  e x y ;  e 2  x  e
1 2
The common decision is such therefore:
x
3
x
y  С 1 e  С 2 e .

5.1.3. The Roots of Characteristic Equation are Real
multiple

Let us take LHDE of the 2-nd order with constant
coefficients
   y a 1 y   a 2 y  0 . (5.6)

We make characteristic Equation

 2  a   a  0 (5.7)
1 2

If left part (5.7) is an exact square, the scolded equations
coincide in this case:    . Then it is possible to write down
1 2
only one partial decision equation (5.6) y  e 1  x . Second
1
decision it is impossible to write down, e 2  x as, y  e 2  x that is
1
decision will be linearly dependent.
Second partial decision linearly independent with the first,
we will search in a kind:
y  e 2  x ,
1
where (xu ) – unknown function which is needed to define. For
this purpose we will put y in equation (5.6). Differentiating we
2
find:

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