  We study quasimodular forms of depth $\leq4$ and determine under which
conditions they occur as solutions of modular differential equations.
Furthermore, we study which modular differential equations have quasimodular
solutions. We use these results to investigate extremal quasimodular forms as
introduced by M. Kaneko and M. Koike further. Especially, we prove a conjecture
stated by these authors concerning the divisors of the denominators occurring
in their Fourier expansion.
