All asymptotic expansions of solutions to the equation P6 in the case $a\cdot b=0$

A. D. Bruno, I. V. Goryuchkina


Abstract: Here we consider the equation P6 for $x\to0$ and $a=0$. In that, we obtained 4 base families of expansions of its solutions different from base families of expansions of solutions for $a\cdot b\ne0$. Altogether for $x\to0$ and $a=0$ there exist 11 families of expansions of solutions involving expansions of four types: power, power-logarithmic, complicated and exotic. Most of them are new. Cases $a=0$, $x\to\infty$ and $b=0$ are obtained from the case $a=0$, $x\to0$ by means of reflections of symmetries of the equation. For $a=b=0$ there are absent expansions of solutions different from expansions of solutions for $a\cdot b\ne0$, $a=0$ and $b=0$. The last paragraph is dedicated to comparison of obtained results with known results.

UDC: 517.91