Invariant sums of products of differentials

Based on the method proposed for solving the so-called $(r, s)$-systems of linear equations, it is proved 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. 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.