Introduction

In 1967, S. Ya. Bartashevich [1] investigated certain classes of nonlinear differential equations. Building upon these results, we consider the following general forms of second-order differential equations:
$$\begin{aligned} y'' + my' + f(x)y + g(x)y^n &= e(x) \end{aligned} \tag{1}$$
$$\begin{aligned} y'' + (m - 1) y' + f(x) y y' + g(x) y^n &= e(x) \end{aligned} \tag{2}$$
$$\begin{aligned} [yy' + (m - 1) y^2 + f(x) y y'] + g(x) y^n &= e(x) \end{aligned} \tag{3}$$

Equations of this type frequently appear in various physical and technical applications [2–6]. Following the methodology in [1], we apply the transformations $x = \alpha(t)$ and $y = \beta(t)z(t)$ to reduce equations (1)–(3) to autonomous or simpler forms, such as:
$$\begin{aligned} z'' + F(t) z' + \Phi(t) z + \Psi(t) z^n &= 0 \end{aligned} \tag{4}$$
$$\begin{aligned} z'' + mz' + F(t) z' + \Phi(t) z^n &= \Psi(t) \end{aligned} \tag{5}$$
$$\begin{aligned} [zz' + (m - 1) z^2 + F(t) z z'] + F(t) z^n &= \Phi(t) \end{aligned} \tag{6}$$

By analyzing the coefficients $\Phi(t)$ and $\Psi(t)$, we can establish the integrability conditions for equations (1)–(3). For instance, the relationship between the coefficients of equation (1) and its transformed counterpart (5) can be expressed through the auxiliary functions $p$ and $q$, where $p = 1$ and $q = m$.

Transformation and Integrability Conditions

For equation (2), assuming the transformation parameters $\alpha$ and $\beta$ satisfy specific constraints, we can derive the functional form of $f(x)$ that allows for exact solutions. When $e(x) = 0$, the relationship between the coefficients $f(x)$ and $g(x)$ is governed by the following differential relations:
$$\begin{aligned} u = \frac{f'(x)}{f(x)}; \quad \frac{F'(t)}{F(t)} \end{aligned} \tag{13}$$
$$\begin{aligned} \frac{g(x)}{[f(x)]^{n+3}} = \frac{\Phi(t)}{[F(t)]^{n+3}} \end{aligned} \tag{15}$$

Substituting these into the original equations, we obtain a system of conditions that define the class of integrable functions $f(x)$ and $g(x)$. Specifically, for equation (3), the transformation leads to:
$$\begin{aligned} \int f(x) dx = \int F(t) dt \end{aligned} \tag{16}$$
The general solution can then be expressed in terms of the transformed variable $z(t)$, which satisfies a simpler autonomous equation.

Special Cases and Applications

We further examine the case where $F(t) = a$ and $\Psi(t) = b$ are constants. As noted in [8], if we introduce the substitution $z' = \xi$, the problem reduces to a first-order equation:
$$\begin{aligned} \xi' + m\xi^2 + F(t)\xi + F(t) = 0 \end{aligned} \tag{23}$$
Under the condition $4ma - b^2 = 0$, the solution simplifies significantly. Using the relations (13), (20), and (21), we can determine the explicit forms of $f(x)$ and $g(x)$ that satisfy the integrability criteria. The resulting general solution for $y(x)$ is then given by:
$$\begin{aligned} z = C_2(t - C_1) e^{-bt/2m} \end{aligned} \tag{28}$$
where the relationship between $x$ and $t$ is defined by the integral of $f(x)$.

These results extend the known classes of integrable second-order nonlinear equations and provide a systematic framework for solving boundary value problems in mathematical physics where such structures arise.

References

1. Bandić, I. "Sur une classe d'équations différentielles non linéaires du deuxième ordre." ZAMM - Journal of Applied Mathematics and Mechanics, 43, 429, 1963.
2. Bogolyubov, N. N., and Mitropolsky, Yu. A. Asymptotic Methods in the Theory of Non-linear Oscillations. Moscow, 1964.
3. Krylov, N. M., and Bogolyubov, N. N. Introduction to Non-linear Mechanics. Kiev, 1934.
4. Kamke, E. Handbook of Ordinary Differential Equations. Vol. 1. Moscow, 1958.
5. Bartashevich, S. Ya. Differential Equations, 15, 107, 1945.
6. Bandić, I. C. R. Acad. Sci. Paris, 260, 6269, 1965.
7. Gradshteyn, I. S., and Ryzhik, I. M. Table of Integrals, Series, and Products. Moscow, 1962.
8. [Additional Reference], 1965.