This report contains a survey on recent results concerning solutions of the second order nonlinear differential equation $x'' + a(t)|x|^\gamma \operatorname{sgn} x = 0$, $\gamma > 0$, where $a(t)$ is continuous and nonnegative. Specifically, the questions concerning continuability, boundedness, stability, oscillation, asymptotic growth and boundary value problems are discussed. An extensive list of related references is also included.

On the Generalized Emden–Fowler Equation

The existence of a Lagrangian description for the second-order Riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized Riccati equation. The Lagrangians are non-natural and the forces are not derivable from a potential. The constant value E of a preserved energy function can be used as an appropriate parameter for characterizing the behavior of the solutions of these two systems. In the second part the existence of two-dimensional versions endowed with superintegrability is proved. The explicit expressions of the additional integrals are obtained in both cases. Finally it is proved that the orbits of the second system, that represents a nonlinear oscillator, can be considered as nonlinear Lissajous figures

Lagrangian formalism for nonlinear second-order Riccati systems: One-dimensional integrability and two-dimensional superintegrability

Hille and Nehari type criteria for third-order dynamic equations

On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations

Comparison theorems for linear dynamic equations on time scales

https://msp.org/pjm/1976/64-2/pjm-v64-n2-p05-s.pdf

Existence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations

/pdf/oscillation-theorems-for-second-order-linear-differential-3v1goj3ejf.pdf

Oscillation theorems for second order linear differential equations.

/pdf/integral-comparison-theorems-for-third-order-linear-3ifqaetilu.pdf

Integral comparison theorems for third order linear differential equations

/pdf/nonoscillatory-solutions-of-second-order-nonlinear-2m7k2hgzqs.pdf

Nonoscillatory solutions of second order nonlinear differential equations

/pdf/comparison-theorems-for-second-order-operator-valued-linear-33hktvrhd0.pdf

Lynn Harry Erbe

Papers

Existence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations

Oscillation theorems for second order linear differential equations.

Integral comparison theorems for third order linear differential equations

Nonoscillatory solutions of second order nonlinear differential equations

Comparison theorems for second-order operator-valued linear differential equations