Internal problem ID [6681]
Book: Second order enumerated odes
Section: section 1
Problem number: 48.
ODE order: 2.
ODE degree: 4.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y \left (y^{\prime \prime }\right )^{4}+\left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 21.234 (sec). Leaf size: 2713
dsolve(y(x)*diff(y(x),x$2)^4+diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = c_{1} \\ y \relax (x ) = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a} \left (\left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {4}{3}}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a} \left (\left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {4}{3}}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a} \left (-\left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {4}{3}}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a} \left (-\left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {4}{3}}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a}^{2} \sqrt {2}}{\sqrt {\textit {\_a} \left (\left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a}^{2} \sqrt {2}}{\sqrt {\textit {\_a} \left (\left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a}^{2} \sqrt {2}}{\sqrt {\textit {\_a} \left (-\left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a}^{2} \sqrt {2}}{\sqrt {\textit {\_a} \left (-\left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a} \left (\left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {4}{3}}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a} \left (\left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {4}{3}}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a} \left (-\left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {4}{3}}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a} \left (-\left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {4}{3}}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {2 \textit {\_a}^{2}}{\sqrt {-2 \textit {\_a} \left (\left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (-1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {2 \textit {\_a}^{2}}{\sqrt {-2 \textit {\_a} \left (\left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (-1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {2 \textit {\_a}^{2}}{\sqrt {-2 \textit {\_a} \left (\left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (-1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {2 \textit {\_a}^{2}}{\sqrt {-2 \textit {\_a} \left (\left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (-1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {2 \textit {\_a}^{2}}{\sqrt {-2 \textit {\_a} \left (-\left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (-1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {2 \textit {\_a}^{2}}{\sqrt {-2 \textit {\_a} \left (-\left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (-1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {2 \textit {\_a}^{2}}{\sqrt {-2 \textit {\_a} \left (-\left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (-1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {2 \textit {\_a}^{2}}{\sqrt {-2 \textit {\_a} \left (-\left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (-1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a}^{2} \sqrt {2}}{\sqrt {\textit {\_a} \left (\left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a}^{2} \sqrt {2}}{\sqrt {\textit {\_a} \left (\left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a}^{2} \sqrt {2}}{\sqrt {\textit {\_a} \left (-\left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a}^{2} \sqrt {2}}{\sqrt {\textit {\_a} \left (-\left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {1}{\RootOf \left (-\ln \left (\textit {\_a} \right )-2 \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_f}}{2 i \left (-\textit {\_f}^{2}\right )^{\frac {1}{4}}+\textit {\_f}^{2}}d \textit {\_f} \right )+c_{1}\right ) \sqrt {\textit {\_a}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-\left (\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-\left (2 i \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (2 i \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a}}{\sqrt {-i \textit {\_a} \left (i \left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a}}{\sqrt {i \textit {\_a} \left (-i \left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a} \sqrt {2}}{\sqrt {\textit {\_a} \left (i \left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}-2 i \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}-2 \sqrt {3}\, \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a} \sqrt {2}}{\sqrt {\textit {\_a} \left (-\left (\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}-2 i \textit {\_a} -2 \sqrt {3}\, \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {\textit {\_a} \sqrt {2}}{\sqrt {\textit {\_a} \left (-\left (2 i \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}-2 i \textit {\_a} +2 \sqrt {3}\, \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-\left (\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-\left (2 i \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (2 i \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a}}{\sqrt {-i \textit {\_a} \left (i \left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a}}{\sqrt {i \textit {\_a} \left (-i \left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {2 \textit {\_a}}{\sqrt {-2 \textit {\_a} \left (-i \left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}-2 \sqrt {3}\, \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {2 \textit {\_a}}{\sqrt {-2 \textit {\_a} \left (-i \left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}-2 \sqrt {3}\, \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {2 \textit {\_a}}{\sqrt {-2 \textit {\_a} \left (-i \left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}+2 \sqrt {3}\, \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {2 \textit {\_a}}{\sqrt {-2 \textit {\_a} \left (-i \left (2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}+2 \sqrt {3}\, \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {2 \textit {\_a}}{\sqrt {-2 \textit {\_a} \left (i \left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}-2 \sqrt {3}\, \textit {\_a} -i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {2 \textit {\_a}}{\sqrt {-2 \textit {\_a} \left (i \left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}-2 \sqrt {3}\, \textit {\_a} -i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {2 \textit {\_a}}{\sqrt {-2 \textit {\_a} \left (-\left (\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}+2 i \textit {\_a} -2 \sqrt {3}\, \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {2 \textit {\_a}}{\sqrt {-2 \textit {\_a} \left (-\left (\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}+2 i \textit {\_a} -2 \sqrt {3}\, \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {2 \textit {\_a}}{\sqrt {-2 \textit {\_a} \left (-\left (2 i \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}+2 i \textit {\_a} +2 \sqrt {3}\, \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {2 \textit {\_a}}{\sqrt {-2 \textit {\_a} \left (-\left (2 i \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}+2 i \textit {\_a} +2 \sqrt {3}\, \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a} \sqrt {2}}{\sqrt {\textit {\_a} \left (i \left (-2 \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}-2 i \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}-2 \sqrt {3}\, \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a} \sqrt {2}}{\sqrt {\textit {\_a} \left (-\left (\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}+2 i \textit {\_a} \right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}-2 i \textit {\_a} -2 \sqrt {3}\, \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\textit {\_a} \sqrt {2}}{\sqrt {\textit {\_a} \left (-\left (2 i \textit {\_a} -\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )^{\frac {1}{3}} \left (i \left (\textit {\_a} c_{1}\right )^{\frac {1}{4}} \sqrt {3}-2 i \textit {\_a} +2 \sqrt {3}\, \textit {\_a} +\left (\textit {\_a} c_{1}\right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {1}{\RootOf \left (-\ln \left (\textit {\_a} \right )-2 \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_f}}{\textit {\_f}^{2}+2 \left (-\textit {\_f}^{2}\right )^{\frac {1}{4}}}d \textit {\_f} \right )+c_{1}\right ) \sqrt {\textit {\_a}}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.729 (sec). Leaf size: 405
DSolve[y[x]*y''[x]^4+y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \left (1-\frac {\left (\frac {2}{3}+\frac {2 i}{3}\right ) \sqrt {2} \text {$\#$1}^{3/4}}{c_1}\right ){}^{2/3} \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {7}{3};\frac {\left (\frac {2}{3}+\frac {2 i}{3}\right ) \sqrt {2} \text {$\#$1}^{3/4}}{c_1}\right )}{\left (\frac {3 c_1}{2}-(1+i) \sqrt {2} \text {$\#$1}^{3/4}\right ){}^{2/3}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \left (1-\frac {\left (\frac {2}{3}-\frac {2 i}{3}\right ) \sqrt {2} \text {$\#$1}^{3/4}}{c_1}\right ){}^{2/3} \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {7}{3};\frac {\left (\frac {2}{3}-\frac {2 i}{3}\right ) \sqrt {2} \text {$\#$1}^{3/4}}{c_1}\right )}{\left (\frac {3 c_1}{2}-(1-i) \sqrt {2} \text {$\#$1}^{3/4}\right ){}^{2/3}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \left (1+\frac {\left (\frac {2}{3}-\frac {2 i}{3}\right ) \sqrt {2} \text {$\#$1}^{3/4}}{c_1}\right ){}^{2/3} \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {7}{3};-\frac {\left (\frac {2}{3}-\frac {2 i}{3}\right ) \sqrt {2} \text {$\#$1}^{3/4}}{c_1}\right )}{\left ((1-i) \sqrt {2} \text {$\#$1}^{3/4}+\frac {3 c_1}{2}\right ){}^{2/3}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \left (1+\frac {\left (\frac {2}{3}+\frac {2 i}{3}\right ) \sqrt {2} \text {$\#$1}^{3/4}}{c_1}\right ){}^{2/3} \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {7}{3};-\frac {\left (\frac {2}{3}+\frac {2 i}{3}\right ) \sqrt {2} \text {$\#$1}^{3/4}}{c_1}\right )}{\left ((1+i) \sqrt {2} \text {$\#$1}^{3/4}+\frac {3 c_1}{2}\right ){}^{2/3}}\&\right ][x+c_2] \\ \end{align*}