Question

Solve the de y"-3y'=sin2x

Answer 1

When it comes to second order differential equations, you have to find the Complementary Function ($y_{C.F.}$) and often times the Particular Solution ($y_{P}$) and the sum of these two would give the General Solution.

Now, for y'' - 3y' = sin2x,
   
$y_{C.F.}$ = A + Be³ˣ
$y_{P}$ = $-\frac{1}{3}$ sin2x + $\frac{3}{26}$ cos2x

Since General Solution = $y_{C.F.}$  +  $y_{P}$
then  General Solution  is y = A + Be³ˣ $-\frac{1}{3}$ sin2x + $\frac{3}{26}$ 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.