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:

(D² – 1)x + (D² – D)y = –2 sin(t)

(D² + D)x + D²y = 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?