Internal problem ID [7437]
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]]
\[ \boxed {y {y^{\prime \prime }}^{4}+{y^{\prime }}^{2}=0} \]
✓ Solution by Maple
Time used: 0.157 (sec). Leaf size: 2926
dsolve(y(x)*diff(y(x),x$2)^4+diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} -\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) {\left (\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (1+i \sqrt {3}\right ) \textit {\_a}^{3} \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\left (i-\sqrt {3}\right ) \textit {\_a}^{3} {\left (\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {-2 \left (1+i \sqrt {3}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (\textit {\_a} +\frac {\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}}{2}\right ) \textit {\_a}^{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} -\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) {\left (\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\left (1-i \sqrt {3}\right ) \textit {\_a}^{3} \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\left (1-i \sqrt {3}\right ) \textit {\_a}^{3} \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} {\left (\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (\sqrt {3}+i\right ) \left (\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} {\left (\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (\sqrt {3}+i\right ) \left (\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (1+i \sqrt {3}\right ) \textit {\_a}^{3} \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\left (i-\sqrt {3}\right ) \textit {\_a}^{3} {\left (\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {-2 \left (1+i \sqrt {3}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (\textit {\_a} +\frac {\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}}{2}\right ) \textit {\_a}^{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\operatorname {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 \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (2 i \textit {\_a} -\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {-i \textit {\_a} {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {i \textit {\_a} {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (i-\sqrt {3}\right ) \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (\sqrt {3}+i\right ) \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \textit {\_a} \left (\sqrt {3}+i\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \textit {\_a} \left (-i+\sqrt {3}\right ) \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (2 i \textit {\_a} -\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {-i \textit {\_a} {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {i \textit {\_a} {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (-i+\sqrt {3}\right ) \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (-i+\sqrt {3}\right ) \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {-\left (\sqrt {3}+i\right ) \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {-\left (\sqrt {3}+i\right ) \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (i-\sqrt {3}\right ) \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (i-\sqrt {3}\right ) \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {-\left (\sqrt {3}+i\right ) \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {-\left (\sqrt {3}+i\right ) \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (i-\sqrt {3}\right ) \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (\sqrt {3}+i\right ) \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \textit {\_a} \left (\sqrt {3}+i\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \textit {\_a} \left (-i+\sqrt {3}\right ) \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\operatorname {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: 4.322 (sec). Leaf size: 1237
DSolve[y[x]*y''[x]^4+y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
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