Root Parametrized Differential Equations for the classical groups

Let $C⟨t_1,\dots t_l⟩$ be the differential field generated by $l$ differential indeterminates $\mathit{t}=(t_1,\dots ,t_l)$ over an algebraically closed field $C$ of characteristic zero. We develop a lower bound criterion for the differential Galois group $G(C)$ of a matrix parameter differential equation $\mathrm{\partial }(\mathit{y})=A(\mathit{t})\mathit{y}$ over $C \langle t_1, \dots t_l\rangle$ and we prove that every connected linear algebraic group is the Galois group of a linear parameter differential equation over $C\langle t_1 \rangle$. As a second application we compute explicit and nice linear parameter differential equations over $C⟨t_1,\dots ,t_l⟩$ for the groups ${\mathrm{S}\mathrm{L}}_{l+1}(C)$, ${\mathrm{S}\mathrm{P}}_{2l}(C)$, ${\mathrm{S}\mathrm{O}}_{2l+1}(C)$, ${\mathrm{S}\mathrm{O}}_{2l}(C)$, i.e. for the classical groups of type $A_l$, $B_l$, $C_l$, $D_l$, and for ${\mathrm{G}}_2$ (here $l=2$).