Question

(D^2+4D+4)y=sin2x find the answer in C.F and P.I format

Answer 1

Notice,

(D2+4)y=sin2x

complementary function (C.F.) is obtained by putting D2+4=0⟹D=±2i

∴C.F.=C1cos2x+C2sin2x

Now, particular integral (P.I.) is found out as follows

P.I.=sin2xD2+4

=1−cos2x2D2+4

=12(1−cos2xD2+4)

=12(1D2+4−cos2xD2+4)

=12(e0⋅xD2+4−xcos2x2D)

=12(102+4−x21D(cos2x))

=12(14−x2sin2x2)

=18−xsin2x8

∴y=CF+PI

y=C1cos2x+C2sin2x+18−xsin2x8

y=C1cos2x+C2sin2x+1−xsin2x8

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