We give a new proof of Jouanolou’s theorem about non-existence of algebraic solutions to the system $ẋ=z^s,ẏ=x^s,ż=y^s$. We also present some generalizations of the results of Darboux and Jouanolou about algebraic Pfaff forms with algebraic solutions.

We study oscillatory properties of solutions of the Emden-Fowler type differential equation $$u^{\left(n\right)}\left(t\right)+p\left(t\right){\left|u\left(\sigma \left(t\right)\right)\right|}^{\lambda }signu\left(\sigma \left(t\right)\right)=0,$$ where $0<\lambda <1$, $p\in L_{\mathrm{loc}}({ℝ}_{+};ℝ)$, $\sigma \in C({ℝ}_{+};{ℝ}_{+})$ and $\sigma \left(t\right)\ge t$ for $t\in {ℝ}_{+}$. Sufficient (necessary and sufficient) conditions of new type for oscillation of solutions of the above equation are established. Some results given in this paper generalize the results obtained in the paper by Kiguradze and Stavroulakis (1998).

For the equation $$y^{\left(n\right)}+{\left|y\right|}^k\mathrm{sgn}y=0,k>1,n=3,4,$$ existence of oscillatory solutions $$y={(x^{*}-x)}^{-\alpha }h(log(x^{*}-x)),\alpha =\frac{n}{k-1},x<x^{*},$$ is proved, where $x^{*}$ is an arbitrary point and $h$ is a periodic non-constant function on $ℝ$. The result on existence of such solutions with a positive periodic non-constant function $h$ on $ℝ$ is formulated for the equation $$y^{\left(n\right)}={\left|y\right|}^k\mathrm{sgn}y,k>1,n=12,13,14.$$

The author considers the quasilinear differential equations $$\begin{array}{c}{\left(r\left(t\right)\varphi \left(x^{\text{'}}\right)\right)}^{\text{'}}+q\left(t\right)f\left(x\right)=0,t\ge a\\ \multicolumn{2}{c}{\text{and}}\\ {\left(r\left(t\right)\varphi \left(x^{\text{'}}\right)\right)}^{\text{'}}+F(t,x)=\pm g\left(t\right),t\ge a.\end{array}$$ By means of topological tools there are established conditions ensuring the existence of nonnegative asymptotic decaying solutions of these equations.