solve differential d^2y/dx^2+4y=sin^2x equation | find solution of Differential Equation (D^2+4) y= sin^2x

find solution of differential equation d²y/dx²+4y=sin²x
solve differential equation  (D²+4) y=sin²x

Solve Differential Equation d²y/dx²+4y=sin²x
Differential it can be written as (D²+4) y= sin²x

Find complementary function

F(D)=0

D²+4=0                                                                    

D=2i    and D=-2i

Roots are imaginary (a+ib). Now

C.F=e^ax(C1 cos bx +C2 sin bx)…………(1)

a=0,b=2

from equation (1)

C.F.=C1 cos 2x +C2 sin 2x………………….(2)

Find particular integral

P.I.=f(x)/f(D)= sin²x/ (D²+4)=1/(D²+4) .(1-cos 2x)/2

      =1/2 [.1/ (D²+4)] -1/2[[.1/ (D²+4) cos2x]    { case is failure)

      =1/2 .1/4 –[x .1/2D cos 2x]

       =1/8 -  x/2 (sin 2x)/2

  P.I.=    1/8-x/4 sin2x

General solution

y=C.F.+P.I.

y= C1 cos 2x +C2 sin 2x+1/8-x/4 sin2x