On a family of ordinary differential equations integrable in elementary functions (Q2210424)

Let \(D\) be the differentiation operator, let the functions \(y_n(x)\) be defined recursively \[ y_0(x)=\exp(x),\dots, y_n(x)=x^n Dx^{-n}y_{n-1}(x), \; n\ge 1. \] Let \( P\) be any polynomisl of degree \( m\). The author considers the differential equation \[ x^n(Dx^{-1})^nP(D)(x^{-1}D)^nx^ny(x)=ay(x) \tag{1} \] with \(a\neq0\) and proves that (1) has a fundamental system of solutions \(\{y_n(\lambda_\nu x)\}\), where \(\lambda_1, \dots, \lambda_{n+m}\) are the roots of the equation \(\lambda^{2n}P(\lambda)=a\), if all roots are different. The case of a multiple root is also treated.