Preamble

In 1967, in the journal TM III (Vol. 11, No. 517.911), and as previously discussed in \cite{1} and \cite{2}, D. D. Baĭnov investigated the $n$-th order differential equation:
$$y^{(n)} + p_0 y^{(n-1)} + \sum_{l=0}^{n-4} \sum_{s=1}^{n-3-l} p_{ls} y^{(n-2-l-s)} y^{(s)} = 0, \quad (n > 6) \tag{1}$$
where $p_0$ and $p_{ls}$ are constants, and $p_0 \neq 0$. We seek a solution to equation (1) in the form of a Dirichlet series:
$$y = \sum_{k=1}^{\infty} a_k e^{-kbx} \tag{2}$$
where $a_k$ are coefficients to be determined, $b$ is a constant such that $Re(b) > 0$, and the series converges for sufficiently large $x$.

By substituting the series (2) into equation (1), we obtain the following relation for the coefficients:
$$\sum_{k=1}^{\infty} a_k (-kb)^n e^{-kbx} + p_0 \sum_{k=1}^{\infty} a_k (-kb)^{n-1} e^{-kbx} + \sum_{l=0}^{n-4} \sum_{s=1}^{n-3-l} p_{ls} \left( \sum_{r=1}^{\infty} a_r (-rb)^{n-2-l-s} e^{-rbx} \right) \left( \sum_{m=1}^{\infty} a_m (-mb)^s e^{-mbx} \right) = 0$$
Equating the coefficients of $e^{-kbx}$ to zero, we derive the recurrence relations for $a_k$. For $k=1$, we have:
$$a_1 (-b)^n + p_0 a_1 (-b)^{n-1} = 0$$
Since we assume $a_1 \neq 0$, it follows that $b = p_0$. For $k > 1$, the coefficients $a_k$ are determined by:
$$a_k [(-kb)^n + p_0 (-kb)^{n-1}] + \sum_{l=0}^{n-4} \sum_{s=1}^{n-3-l} p_{ls} \sum_{r=1}^{k-1} a_r a_{k-r} (-rb)^{n-2-l-s} (-(k-r)b)^s = 0 \tag{3}$$
From (3), we can express $a_k$ as:
$$a_k = \frac{1}{k^{n-1}(k-1) p_0^n} \sum_{l=0}^{n-4} \sum_{s=1}^{n-3-l} p_{ls} \sum_{r=1}^{k-1} a_r a_{k-r} r^{n-2-l-s} (k-r)^s p_0^{n-2-l} \tag{4}$$
where $k = 2, 3, \dots$.

To ensure the convergence of the series (2), we analyze the growth of the coefficients $a_k$. Let $|a_1| = A$. We aim to show that $|a_k| \leq A^k$ for all $k$. Using the properties of the beta function and the maximum values of the terms in the summation, we consider the integral:
$$\int_0^1 x^{n-2-l-s} (1-x)^s dx = \frac{(n-2-l-s)! s!}{(n-1-l)!}$$
By applying estimates to the summation in (4), we can establish bounds on $|a_k|$. Specifically, we utilize the fact that for $x \in [0, 1]$, the function $f(x) = x^k(1-x)^m$ reaches its maximum at $x = \frac{k}{k+m}$.

Following the methodology of D. D. Baĭnov, we evaluate the sum:
$$\sum_{l=0}^{n-4} \sum_{s=1}^{n-3-l} \frac{(n-2-l-s)! s!}{(n-1)!} \left( \frac{n-2-l-s}{n-2-l} \right)^{n-2-l-s} \left( \frac{s}{n-2-l} \right)^s$$
After simplification and applying the condition $n > 6$, it can be shown that if the coefficients of the original equation satisfy the condition:
$$\max(|p_0|, |p_{ls}|) < \frac{(n-1)(n-2)(n-3)}{4(n-2)^2 + (n-3)^2}$$
then the coefficients $|a_k|$ remain bounded such that the Dirichlet series (2) converges for $Re(p_0 x) > \ln A$.

Thus, we have demonstrated that under the specified constraints on the parameters $p_0$ and $p_{ls}$, the nonlinear differential equation (1) admits a solution in the form of a convergent Dirichlet series. This result extends the findings presented in \cite{1} and \cite{2} regarding the existence of analytic solutions for high-order nonlinear equations.

References

1. Baĭnov, D. D. "On a nonlinear differential equation of $n$-th order," Journal of Mathematical Analysis, No. 3, pp. 327–330, 1965.
2. Baĭnov, D. D. "Convergence of Dirichlet series solutions," Reports of the Bulgarian Academy of Sciences, Vol. 2, No. 6, pp. 853–854, 1966.