Introduction

This section examines a class of differential equations and their asymptotic properties, as presented in the work submitted to Moscow State University in May 1967. We consider a system of the form:

$$u' = U_0(v)u + \sum U_n(v)u^{1+\alpha_n} = U(u, v)$$

where $u$ and $v$ are variables, and the coefficients $U_s(v)$ are defined by the series:

$$U_s(v) = \sum U_{s,j} e^{p_s v}$$

In this context, $p$ is a constant, and the functions $U(u, v)$ and $U_s(v)$ are assumed to be analytic. We further assume the conditions $p_{s,j} < p_{s, j+1}$ and $p_{s,0} = 0$ for $s \geq 1$ and $j \geq 1$. For the initial values $u \approx u_0$ and $v \approx v_0$, we define the characteristic term $U_{0,0}(v) \approx v_0$. The expansion coefficients are given by:

$$U_n(v) = \sum u_{n,s}(v) e^{b_{n,s} v}$$

where $b_n$ and $b_{n,s}$ are constants satisfying $b_n < b_{n+1}$ and $b_{n,s} < b_{n, s+1}$. It is established that $u_{n,0} = 0$ for $n > 1$ and $s > 0$.

By applying a transformation to the original system (1), we can reduce it to the following form:

$$z' = 1 + \sum L_n(v) z^{n}$$

where the functions $L_n(v)$ are related to the original coefficients $U_n(v)$ through the integral:

$$L_n(v) = U_n(v) \exp\left( \alpha_n \int U_0(\eta) d\eta \right)$$

Based on this formulation, we consider two primary cases. First, we analyze the behavior of the system when the constant $c'$ is fixed. Second, we examine the conditions under which the solution $u(v)$ can be represented as a series:

$$u = c + \sum u_n(v) u^n$$

where $c$ and $b_n$ are positive constants. For the system described in (1), we assume the sequence of exponents satisfies $b_n < b_{n+1}$ for $n \geq -1$. The functions $U_n(v)$ are defined according to the properties established in (2). This analytical framework was formally presented at Moscow State University by M. V. [Name truncated] on May 16, 1967.