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Thus, the partial decision of LNDE has a kind
1 2 1 1 2
y *  x  x   ( x ) 1 .
2 2 2

Common decision of given LNDE:
~ x 1 2
y  y  y *  e (C cos  Cx sin x )  ( x ) 1 .
1 2
2

Example 6.5 To find a common decision of equation
x
2
2 y   y   y  4 xe .

 We have LNDE with constant coefficients with right part
of the special kind. Proper to him LHDE
2  y    y y  0.
We make characteristic equation
2 2    1  0,
1
are scolded which   , 1    .
1 2
2
Common decision LHDE:
x

~ C e  C e .
x
y 
2
1 2
We shall search the partial decision LNDE by the method of
selection according to the type of right part:
* y  e 2x (Ax  ) B ,
x
2
because right part of given LNDE f ( x)  4 xe answers a case

x
f ( x)  Q ( x) e at  s , 1  2 , thus   2 is not the root of
s
characteristic equation.
We find * yy , * and put *, yy , * y * in LNDE. As a
result we will get an identity. Again for comfort of calculations
we write down expressions for *, yy , * y * in separate lines,
and to the left after a vertical hyphen - coefficients, with which
they enter to left part LNDE. We multiply *, yy , * y * on the
proper coefficients, add, erect like term and have an identity
which is written down than horizontal hyphen below. Thus
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