The characteristic equation k2+4=0 has roots k1=2i and k2=−2i. Therefore, the general solution of the differential equation is of the form:
y=C1cos2x+C2sin2x+yp,
where yp=acos5x+bsin5x+cx+d. It follows that
yp′=−5asin5x+5bcos5x+c,yp′′=−25acos5x−25bsin5x
Then
−25acos5x−25bsin5x+4(acos5x+bsin5x+cx+d)=sin5x+x−1.
We get the following equation
−21acos5x−21bsin5x+4cx+4d=sin5x+x−1,
and hence
−21a=0, −21b=1, 4c=1, 4d=−1.
It follows that a=0, b=−211, c=41, d=−41.
We conclude that the general solution of the differential equation is of the form
y=C1cos2x+C2sin2x−211sin5x+41x−41.
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