58.1.48 problem 48

Internal problem ID [9119]
Book : Second order enumerated odes
Section : section 1
Problem number : 48
Date solved : Monday, January 27, 2025 at 05:42:49 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y {y^{\prime \prime }}^{4}+{y^{\prime }}^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 13.932 (sec). Leaf size: 2829

dsolve(y(x)*diff(y(x),x$2)^4+diff(y(x),x)^2=0,y(x), singsol=all)
 

\begin{align*} y &= c_{1} \\ y &= 0 \\ \int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} -\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}-2 \textit {\_a} \right ) {\left (\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{{1}/{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{{1}/{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{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 )^{{1}/{4}}\right )^{{1}/{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\left (i-\sqrt {3}\right ) \textit {\_a}^{3} {\left (\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \left (\left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {-2 \textit {\_a}^{3} \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) \left (\textit {\_a} +\frac {\left (c_{1} \textit {\_a} \right )^{{1}/{4}}}{2}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} -\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ -\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}-2 \textit {\_a} \right ) {\left (\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{{1}/{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ -\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ -\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{{1}/{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\left (1-i \sqrt {3}\right ) \textit {\_a}^{3} \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\left (1-i \sqrt {3}\right ) \textit {\_a}^{3} \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} {\left (\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \left (\sqrt {3}+i\right ) \left (\left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} {\left (\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \left (\sqrt {3}+i\right ) \left (\left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{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 )^{{1}/{4}}\right )^{{1}/{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\left (i-\sqrt {3}\right ) \textit {\_a}^{3} {\left (\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \left (\left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {-2 \textit {\_a}^{3} \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) \left (\textit {\_a} +\frac {\left (c_{1} \textit {\_a} \right )^{{1}/{4}}}{2}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \int _{}^{y}\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 )^{{1}/{4}}+\textit {\_f}^{2}}d \textit {\_f} \right )+c_{1} \right ) \sqrt {\textit {\_a}}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (2 i \textit {\_a} -\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (\left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y}\frac {\textit {\_a}}{\sqrt {-i \textit {\_a} {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y}\frac {\textit {\_a}}{\sqrt {i \textit {\_a} {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (-i+\sqrt {3}\right ) \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\left (\sqrt {3}+i\right ) \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}-2 \textit {\_a} \right ) \textit {\_a} \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (\sqrt {3}+i\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (2 i \textit {\_a} -\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ -\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (\left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} &= 0 \\ -\int _{}^{y}\frac {\textit {\_a}}{\sqrt {-i \textit {\_a} {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ -\int _{}^{y}\frac {\textit {\_a}}{\sqrt {i \textit {\_a} {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {-\left (\sqrt {3}+i\right ) \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) \textit {\_a} \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {-\left (\sqrt {3}+i\right ) \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right ) \textit {\_a} \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\left (-i+\sqrt {3}\right ) \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\left (-i+\sqrt {3}\right ) \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {-\left (\sqrt {3}+i\right ) \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {-\left (\sqrt {3}+i\right ) \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\left (i-\sqrt {3}\right ) \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\left (i-\sqrt {3}\right ) \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\left (i-\sqrt {3}\right ) \textit {\_a} \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}-2 \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\left (i-\sqrt {3}\right ) \textit {\_a} \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}-2 \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (-i+\sqrt {3}\right ) \left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}+2 \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\left (\sqrt {3}+i\right ) \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\left (i \left (c_{1} \textit {\_a} \right )^{{1}/{4}}-2 \textit {\_a} \right ) \textit {\_a} \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{{1}/{4}}\right )^{{1}/{3}} \left (\sqrt {3}+i\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \int _{}^{y}\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 )^{{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: 3.846 (sec). Leaf size: 1237

DSolve[y[x]*D[y[x],{x,2}]^4+D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

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