Invariant differential polynomials

Based on the method proposed in the article for solving the so-called $(r,s)$-systems of linear equations proven that the orders of homogeneous invariant differential operators $n$ of smooth real functions of one variable take values from $n$ to $\frac{n(n+1)}{2}$, and the dimension of the space of all such operators does not exceed $n!$. A classification of invariant differential operators of order $n+s$ is obtained for $s=1,2,3,4$, and for $n=4$ for all orders from 4 to 10. The only, up to factors, homogeneous invariant differential operators of the smallest order $n$ and the largest order $\frac{n(n+1)}{2}$ are given, respectively, by the product of the $n$ first differentials ($s=0$ ) and the Wronskian ($s=(n-1)n/2$). The existence of nonzero homogeneous invariant differential operators of order $n+s$ for $s<\frac{1+\sqrt{5}}{2}(n-1)$ is proved.