<s>[INST] Nonlinear ordinary differential equation (Elsgolts) Please, help me to solve the following non-linear ODEs:
\begin{align}
\tag 1 y &= (y')^4 -(y')^3 -2 \\
\tag 2 y' &= \dfrac{y}{x+ y^3}
\end{align}
Thanks.
 [/INST] $$ y = y'^4 - y'^3 - 2 $$ 
$$ y' = p \Rightarrow y = p^4 - p^3 - 2 $$
$$ \Rightarrow y' = p = p^2(4p-3)\frac{dp}{dx} $$
$$ \Rightarrow p(4p-3) \ dp = dx \Rightarrow \int 4p^2 - 3p \ dp = \int dx $$
$$ \Rightarrow \frac{4p^3}{3} - \frac{3p^2}{2} + c_1 = x $$
$$  \Rightarrow  \left\{\begin{matrix}
y = p^4 - p^3 - 2 \\ 
\frac{4p^3}{3} - \frac{3p^2}{2} + c_1 = x  \\
\end{matrix}\right. $$
</s>