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)` =
`-(1)/(3)` sin2x +
`(3)/(26)` cos2x

Since General Solution =
`y_(C.F.)` +
`y_(P)`
then General Solution is y = A + Be³ˣ
`-(1)/(3)` sin2x +
`(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.