Yes, C is a constant. Looking back at my notes, it looks like what I did was to set q(t) = B+C*t, then di/dt = i*A/q, and since dq/dt = C, (di/dt)*(dt/dq) = di/dq = i*A/(C*q) and I then integrated (1/i)*di = (A/C)*(1/q)*dq to find i(q) and made some substitutions to find i(t).
But back to...