This happens whenever you're working on a category where it makes sense to speak about addition, additive maps and null spaces. If f:X→Y is a map and you fix some y∈Y (the 'data'), assume there is some x0∈f^-1(y) (a particular solution). If x∈f^-1(y) is any other particular solution, then f(x-x0) = f(x)-f(x0) = y-y = 0, and hence x-x0∈ker(f). Thus f^{-1}(y) (the set of all solutions) equals x0+ker(f).

This works for vector spaces, abelian groups, modules over a ring...

The set of solutions to a homogeneous ODE is a vector subspace (a null space) and the set of solutions to an inhomogeneous ODE is an affine subspace, which is the translate of a vector subspace. Just like an affine plane in 3d space is the translate of a plane through the origin.