Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 17 / b / i / Solution

Yes. The differential operator $\mathcal{L}=2y{\partial }_x-x{\partial }_y$ is linear, even though its coefficients depend on position. For constants $A,B$,If each homogeneous right-hand side is zero, so is their combination. Hence the superposition principle holds.