Invariance of some forms of functions (Q1190051)

Differential operators of the kind \(\Phi\xi=k\xi''/\xi+l\xi'''/\xi'+p(\xi'/\xi)^ 2+q(\xi''/\xi')^ 2)\) (where \(k,l,p,q\) are constants, \(\xi=\xi(x))\) satisfying the identity \(\Phi(f\circ\varphi)=\Phi\varphi\) for every function \(\varphi=\varphi(x)\) and appropriate \(f=f(x)\) are determined. In general \(f=A(x^ B+C)^{1/B}\) where \(A,B,C\) are constants depending on \(k,l,p,q\) (which cannot be arbitrary). The particular case \(m=-1\) yields the familiar Schwarz derivative.