  In these lectures, we discuss two approaches to studying orbit spaces of
algebraic Lie groups. Due to algebraic approach orbit space, or quotient, is an
algebraic manifold, while from the differential viewpoint a quotient is a
differential equation. The main goal of these lectures is to show that the
differential approach gives us a better understanding of structure of
invariants and orbit spaces. We illustrate this on classical equivalence
problems, such as $\mathrm{SL}$ - classification of binary and ternary forms,
and affine classification of algebraic plane curves.
