Question

(D -2) x + Dy = 10sin2t

Dx + (D+2)y =0

Answer 1

From

$$(D-2)x +Dy =10 \sin t$$

We have:

$$Dx-2x+Dy=10 \sin t \quad \cdots (i)$$

Also from

$$Dx+(D+2)y =0$$

We have:

$$Dx+Dy+2y=0\\ Dy=-Dx-2y \quad \cdots (ii)$$

Put (ii) in (i):

$Dx-2x-Dx-2y=10 \sin t\\ -2x-2y = 10 \sin t\\ -2(x+y)=10 \sin t\\ x+y=-5 \sin t\\$

Thus

$$y=-x-5 \sin t$$

is the solution