Analytical solutions to differential equations with singularities and shifts (Q5929644)

Here, the authors study first-order differential equations with higher-order singularities and shifts and derive necessary and sufficient conditions for their solvability. The formulae derived in the paper are applied to the Poincaré problem. Under certain conditions the authors consider the equation \[ z^n\phi'(z)- \alpha(z)\phi(\beta(z))= f(z),\quad z\in D^+,\tag{1} \] where \(\alpha(z)\), \(\beta(z)\) and \(f(z)\) are functions analytic in \(D^+\), \(D^+\) is the unit disk \(|z|< 1\), \(n\geq 2\) is a natural number. The paper consists of six sections. In section 1, the authors prove their main theorem. In section 2, they propose a simple method to solve (1). In section 3 and 5 they present an explicit formula for the solution to (1) and some of its versions. In section 4, they investigate the case \(\beta'(0)= 0\). In section 6, they apply their results to the Laplace equation in the Poincaré problem.