When it comes to second order differential equations, you have to find the Complementary Function ([tex] y_{C.F.}[/tex]) and often times the Particular Solution ([tex] y_{P}[/tex]) and the sum of these two would give the General Solution.
Now, for y'' - 3y' = sin2x,
[tex] y_{C.F.}[/tex] = A + Be³ˣ
[tex] y_{P}[/tex] = [tex] -\frac{1}{3} [/tex] sin2x + [tex] \frac{3}{26} [/tex] cos2x
Since General Solution = [tex] y_{C.F.}[/tex] + [tex] y_{P}[/tex]
then General Solution is y = A + Be³ˣ [tex] -\frac{1}{3} [/tex] sin2x + [tex] \frac{3}{26} [/tex] cos2x
to remove the fraction you can multiply through by 26
⇒ General Solution is also 26y = 26A + 26Be³ˣ - 2sin2x + 3cos2x
Attached below is a picture that shows you exactly how I arrived at my answer.
