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 1 * y  e 2x (Ax  ) B

 1 * y  2e 2x (Ax  B ) e 2x A  e 2x 2 ( Ax  2B  ) A
2 * y   2e 2x 2 ( Ax  2B  A ) e 2x 2A  e 2x 4 ( Ax  4B  4A )
x
2
x
2
2 y *   y *   y*  e ( 5 Ax 5 B 7 A)  4 xe

x
2
let us simplify on e and equating coefficients at identical
degrees x , we get
x 1 5 A 4 4 28
, from where A  , B   .
x 0 5 B 7 A 0 5 25

Consequently, the partial decision LNDE has a kind
 4 28 
* y  e 2x  x   .
 5 25 

Common decision of LNDE is
x
~ x  2 2x  4 28 
y  y  *y  C e  C e  e  x   . 
1 2
 5 25 

Example 6.6 To find the common decision of equation
x
y   2 y   y  xe .

 We have LNDE with constant coefficients with right part
of the special kind. According to LHDE we get

   y 2  y  y  0.

We make characteristic equation  2  2  1  0, and
    1 .
1 2
~
x
Common decision of LHDE is y  C (  C x) e .
1 2

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