On the solution to a singular Cauchy problem for a first-order differential equation unsolved with respect to the derivative of unknown function (Q2768827)

The author considers the following singular Cauchy problem NEWLINE\[NEWLINE\sum_{i+j+k=1}^{m}a_{ijk}t^ix^j{x'}^k+f(t,x,x')=0,\quad \lim_{t\to +0}x(t)=x_0.\tag{1}NEWLINE\]NEWLINE Here, \(f\in C(D\mapsto\mathbb{R})\) with \(D:=\{(t,x,y):t\in (0,\tau)\), \(|x|<\mu t\), \(|y|<\mu \}\), \(\mu >0\), \(\tau >0\), \(f(t,x,y)=o(t^m)\) and \(|f(t,x_1,y_1)-f(t,x_2,y_2)|\leq t\beta(t)(|x_1-x_2|+|y_1-y_2|)\), and \(\beta:(0,\infty) \mapsto(0,\infty), \lim_{t\to +0}\beta(t)=0\). Under the assumption that the equation NEWLINE\[NEWLINE\sum_{k=0}^{m-1}a_{10k}c^k+\sum_{k=0}^{m-1}a_{01k}c^{k+1}=0NEWLINE\]NEWLINE has a multiple root \(c=c_0\) such that \(|c_0|<\mu \) and \(\sum_{k=0}^{m-1}a_{01k}c_0^{k}\neq 0\), the author imposes conditions on \(\beta(t)\) which guarantee the existence of a unique solution to (1) with asymptotic behavior \(x(t)\sim c_0t\), \(x'(t)\sim c_0\), \(t\to 0\). In terms of \(\beta(t)\), he also obtains estimates on \(|x(t)-c_0t|\) and \(|x'(t)-c_0|\).