The letter D is used to denote differentiation of a function of t.
x and y are both functions of t.

Using unimodular row reduction, I want to solve the system:
(D 1)x + (D D)y = 2 sin(t)
(D + D)x + Dy = 0

I have already reduced the system to:
(D+1)x + Dy = 2sin(t)
0x + 0y = cos(t)

I notice that from the second equation, the system is consistent only if cos(t) = 0 in which case y will be a free variable, but how do I proceed from there to determine the solution to the system?