The generating elements of certain Volterra operators connected with third- and fourth-order differential operators

Sufficient conditions are established for $f(x)$ to be the generating function for the Volterra operator which is inverse to the Cauchy operator: $l[y]=y^{(n)}+p_2(x)y^{(n-2)}+\dots+p_n(x)y$, $y(0)=y'(0)=\dots=y^{(n-1)}(0)=0$ ($n=3,4$), when the coefficients $p_i(x)$ are not analytic.