ODE
\[ -2 x \left (x^2+2 y(x)^2\right ) y(x) y'(x)+\left (x^4+y(x)^2 \left (x^2-y(x)^2\right )\right ) y'(x)^2+y(x)^4=0 \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
Homogeneous ODE, \(x^n f\left ( \frac {y}{x} , y' \right )=0\), Solve for \(p\)
Mathematica ✓
cpu = 0.998734 (sec), leaf count = 4936
\[\left \{\left \{y(x)\to -\frac {\sqrt {\frac {x^4-10 \sinh (2 c_1) x^2-2 \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}} x^2+\cosh (4 c_1)+\sinh (4 c_1)-2 \cosh (2 c_1) \left (5 x^2+\sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}\right )+\left (x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}\right ){}^{2/3}-2 \sinh (2 c_1) \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}}{\sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}}}}{\sqrt {3}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {x^4-10 \sinh (2 c_1) x^2-2 \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}} x^2+\cosh (4 c_1)+\sinh (4 c_1)-2 \cosh (2 c_1) \left (5 x^2+\sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}\right )+\left (x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}\right ){}^{2/3}-2 \sinh (2 c_1) \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}}{\sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}}}}{\sqrt {3}}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {i \sqrt {3} x^4-x^4-10 i \sqrt {3} \sinh (2 c_1) x^2+10 \sinh (2 c_1) x^2-4 \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}} x^2+i \left (i+\sqrt {3}\right ) \cosh (4 c_1)+i \sqrt {3} \sinh (4 c_1)-\sinh (4 c_1)-2 \cosh (2 c_1) \left (5 i \left (i+\sqrt {3}\right ) x^2+2 \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}\right )-i \sqrt {3} \left (x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}\right ){}^{2/3}-\left (x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}\right ){}^{2/3}-4 \sinh (2 c_1) \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}}{\sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}}}}{\sqrt {6}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {i \sqrt {3} x^4-x^4-10 i \sqrt {3} \sinh (2 c_1) x^2+10 \sinh (2 c_1) x^2-4 \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}} x^2+i \left (i+\sqrt {3}\right ) \cosh (4 c_1)+i \sqrt {3} \sinh (4 c_1)-\sinh (4 c_1)-2 \cosh (2 c_1) \left (5 i \left (i+\sqrt {3}\right ) x^2+2 \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}\right )-i \sqrt {3} \left (x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}\right ){}^{2/3}-\left (x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}\right ){}^{2/3}-4 \sinh (2 c_1) \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}}{\sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}}}}{\sqrt {6}}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {-i \sqrt {3} x^4-x^4+10 i \sqrt {3} \sinh (2 c_1) x^2+10 \sinh (2 c_1) x^2-4 \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}} x^2+\left (-1-i \sqrt {3}\right ) \cosh (4 c_1)-i \sqrt {3} \sinh (4 c_1)-\sinh (4 c_1)+\cosh (2 c_1) \left (10 \left (1+i \sqrt {3}\right ) x^2-4 \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}\right )+i \sqrt {3} \left (x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}\right ){}^{2/3}-\left (x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}\right ){}^{2/3}-4 \sinh (2 c_1) \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}}{\sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}}}}{\sqrt {6}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {-i \sqrt {3} x^4-x^4+10 i \sqrt {3} \sinh (2 c_1) x^2+10 \sinh (2 c_1) x^2-4 \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}} x^2+\left (-1-i \sqrt {3}\right ) \cosh (4 c_1)-i \sqrt {3} \sinh (4 c_1)-\sinh (4 c_1)+\cosh (2 c_1) \left (10 \left (1+i \sqrt {3}\right ) x^2-4 \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}\right )+i \sqrt {3} \left (x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}\right ){}^{2/3}-\left (x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}\right ){}^{2/3}-4 \sinh (2 c_1) \sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}}{\sqrt [3]{x^6-15 \cosh (2 c_1) x^4-15 \sinh (2 c_1) x^4+39 \cosh (4 c_1) x^2+39 \sinh (4 c_1) x^2+\cosh (6 c_1)+\sinh (6 c_1)+6 \sqrt {3} \sqrt {x^2 \left (-\cosh (6 c_1) x^4-\sinh (6 c_1) x^4+11 \cosh (8 c_1) x^2+11 \sinh (8 c_1) x^2+\cosh (10 c_1)+\sinh (10 c_1)\right )}}}}}{\sqrt {6}}\right \}\right \}\]
Maple ✓
cpu = 6.497 (sec), leaf count = 1001
\[\left [y \left (x \right ) = \frac {\left (-\textit {\_C1}^{3}+18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}{3}-\frac {3 \left (\frac {x^{2}}{3}-\frac {\textit {\_C1}^{2}}{9}\right )}{\left (-\textit {\_C1}^{3}+18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}+\frac {2 \textit {\_C1}}{3}, y \left (x \right ) = \frac {\left (\textit {\_C1}^{3}-18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}{3}-\frac {3 \left (\frac {x^{2}}{3}-\frac {\textit {\_C1}^{2}}{9}\right )}{\left (\textit {\_C1}^{3}-18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}-\frac {2 \textit {\_C1}}{3}, y \left (x \right ) = -\frac {\left (-\textit {\_C1}^{3}+18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}{6}+\frac {\frac {x^{2}}{2}-\frac {\textit {\_C1}^{2}}{6}}{\left (-\textit {\_C1}^{3}+18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}+\frac {2 \textit {\_C1}}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (-\textit {\_C1}^{3}+18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}{3}+\frac {x^{2}-\frac {\textit {\_C1}^{2}}{3}}{\left (-\textit {\_C1}^{3}+18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (-\textit {\_C1}^{3}+18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}{6}+\frac {\frac {x^{2}}{2}-\frac {\textit {\_C1}^{2}}{6}}{\left (-\textit {\_C1}^{3}+18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}+\frac {2 \textit {\_C1}}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (-\textit {\_C1}^{3}+18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}{3}+\frac {x^{2}-\frac {\textit {\_C1}^{2}}{3}}{\left (-\textit {\_C1}^{3}+18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (\textit {\_C1}^{3}-18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}{6}+\frac {\frac {x^{2}}{2}-\frac {\textit {\_C1}^{2}}{6}}{\left (\textit {\_C1}^{3}-18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}-\frac {2 \textit {\_C1}}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (\textit {\_C1}^{3}-18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}{3}+\frac {x^{2}-\frac {\textit {\_C1}^{2}}{3}}{\left (\textit {\_C1}^{3}-18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (\textit {\_C1}^{3}-18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}{6}+\frac {\frac {x^{2}}{2}-\frac {\textit {\_C1}^{2}}{6}}{\left (\textit {\_C1}^{3}-18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}-\frac {2 \textit {\_C1}}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (\textit {\_C1}^{3}-18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}{3}+\frac {x^{2}-\frac {\textit {\_C1}^{2}}{3}}{\left (\textit {\_C1}^{3}-18 x^{2} \textit {\_C1} +3 \sqrt {-3 \textit {\_C1}^{4} x^{2}+33 x^{4} \textit {\_C1}^{2}+3 x^{6}}\right )^{\frac {1}{3}}}\right )}{2}\right ]\] Mathematica raw input
DSolve[y[x]^4 - 2*x*y[x]*(x^2 + 2*y[x]^2)*y'[x] + (x^4 + y[x]^2*(x^2 - y[x]^2))*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[(x^4 + Cosh[4*C[1]] - 10*x^2*Sinh[2*C[1]] + Sinh[4*C[1]] - 2*x^2*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3) - 2*Sinh[2*C[1]]*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3) + (x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(2/3) - 2*Cosh[2*C[1]]*(5*x^2 + (x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3)))/(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3)]/Sqrt[3])}, {y[x] -> Sqrt[(x^4 + Cosh[4*C[1]] - 10*x^2*Sinh[2*C[1]] + Sinh[4*C[1]] - 2*x^2*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3) - 2*Sinh[2*C[1]]*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3) + (x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(2/3) - 2*Cosh[2*C[1]]*(5*x^2 + (x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3)))/(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3)]/Sqrt[3]}, {y[x] -> -(Sqrt[(-x^4 + I*Sqrt[3]*x^4 + I*(I + Sqrt[3])*Cosh[4*C[1]] + 10*x^2*Sinh[2*C[1]] - (10*I)*Sqrt[3]*x^2*Sinh[2*C[1]] - Sinh[4*C[1]] + I*Sqrt[3]*Sinh[4*C[1]] - 4*x^2*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3) - 4*Sinh[2*C[1]]*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3) - (x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(2/3) - I*Sqrt[3]*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(2/3) - 2*Cosh[2*C[1]]*((5*I)*(I + Sqrt[3])*x^2 + 2*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3)))/(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3)]/Sqrt[6])}, {y[x] -> Sqrt[(-x^4 + I*Sqrt[3]*x^4 + I*(I + Sqrt[3])*Cosh[4*C[1]] + 10*x^2*Sinh[2*C[1]] - (10*I)*Sqrt[3]*x^2*Sinh[2*C[1]] - Sinh[4*C[1]] + I*Sqrt[3]*Sinh[4*C[1]] - 4*x^2*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3) - 4*Sinh[2*C[1]]*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3) - (x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(2/3) - I*Sqrt[3]*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(2/3) - 2*Cosh[2*C[1]]*((5*I)*(I + Sqrt[3])*x^2 + 2*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3)))/(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3)]/Sqrt[6]}, {y[x] -> -(Sqrt[(-x^4 - I*Sqrt[3]*x^4 + (-1 - I*Sqrt[3])*Cosh[4*C[1]] + 10*x^2*Sinh[2*C[1]] + (10*I)*Sqrt[3]*x^2*Sinh[2*C[1]] - Sinh[4*C[1]] - I*Sqrt[3]*Sinh[4*C[1]] - 4*x^2*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3) - 4*Sinh[2*C[1]]*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3) - (x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(2/3) + I*Sqrt[3]*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(2/3) + Cosh[2*C[1]]*(10*(1 + I*Sqrt[3])*x^2 - 4*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3)))/(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3)]/Sqrt[6])}, {y[x] -> Sqrt[(-x^4 - I*Sqrt[3]*x^4 + (-1 - I*Sqrt[3])*Cosh[4*C[1]] + 10*x^2*Sinh[2*C[1]] + (10*I)*Sqrt[3]*x^2*Sinh[2*C[1]] - Sinh[4*C[1]] - I*Sqrt[3]*Sinh[4*C[1]] - 4*x^2*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3) - 4*Sinh[2*C[1]]*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3) - (x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(2/3) + I*Sqrt[3]*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(2/3) + Cosh[2*C[1]]*(10*(1 + I*Sqrt[3])*x^2 - 4*(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3)))/(x^6 - 15*x^4*Cosh[2*C[1]] + 39*x^2*Cosh[4*C[1]] + Cosh[6*C[1]] - 15*x^4*Sinh[2*C[1]] + 39*x^2*Sinh[4*C[1]] + Sinh[6*C[1]] + 6*Sqrt[3]*Sqrt[x^2*(-(x^4*Cosh[6*C[1]]) + 11*x^2*Cosh[8*C[1]] + Cosh[10*C[1]] - x^4*Sinh[6*C[1]] + 11*x^2*Sinh[8*C[1]] + Sinh[10*C[1]])])^(1/3)]/Sqrt[6]}}
Maple raw input
dsolve((x^4+(x^2-y(x)^2)*y(x)^2)*diff(y(x),x)^2-2*x*y(x)*(x^2+2*y(x)^2)*diff(y(x),x)+y(x)^4 = 0, y(x))
Maple raw output
[y(x) = 1/3*(-_C1^3+18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)-3*(1/3*x^2-1/9*_C1^2)/(-_C1^3+18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)+2/3*_C1, y(x) = 1/3*(_C1^3-18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)-3*(1/3*x^2-1/9*_C1^2)/(_C1^3-18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)-2/3*_C1, y(x) = -1/6*(-_C1^3+18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)+3/2*(1/3*x^2-1/9*_C1^2)/(-_C1^3+18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)+2/3*_C1-1/2*I*3^(1/2)*(1/3*(-_C1^3+18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)+3*(1/3*x^2-1/9*_C1^2)/(-_C1^3+18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)), y(x) = -1/6*(-_C1^3+18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)+3/2*(1/3*x^2-1/9*_C1^2)/(-_C1^3+18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)+2/3*_C1+1/2*I*3^(1/2)*(1/3*(-_C1^3+18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)+3*(1/3*x^2-1/9*_C1^2)/(-_C1^3+18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)), y(x) = -1/6*(_C1^3-18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)+3/2*(1/3*x^2-1/9*_C1^2)/(_C1^3-18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)-2/3*_C1-1/2*I*3^(1/2)*(1/3*(_C1^3-18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)+3*(1/3*x^2-1/9*_C1^2)/(_C1^3-18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)), y(x) = -1/6*(_C1^3-18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)+3/2*(1/3*x^2-1/9*_C1^2)/(_C1^3-18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)-2/3*_C1+1/2*I*3^(1/2)*(1/3*(_C1^3-18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3)+3*(1/3*x^2-1/9*_C1^2)/(_C1^3-18*x^2*_C1+3*(-3*_C1^4*x^2+33*_C1^2*x^4+3*x^6)^(1/2))^(1/3))]