2.318   ODE No. 318

$\left (3 x y(x)^3-4 x y(x)+y(x)\right ) y'(x)+\left (y(x)^2-2\right ) y(x)^2=0$ Mathematica : cpu = 0.222283 (sec), leaf count = 4284

$\left \{\{y(x)\to 0\},\left \{y(x)\to -\sqrt {\frac {4 \sqrt [3]{2} x^2}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {4 \sqrt [3]{2} x}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {\sqrt [3]{2}}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {2}{3}-\frac {2}{3 x}+\frac {\sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}{3 \sqrt [3]{2} x^2}}\right \},\left \{y(x)\to \sqrt {\frac {4 \sqrt [3]{2} x^2}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {4 \sqrt [3]{2} x}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {\sqrt [3]{2}}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {2}{3}-\frac {2}{3 x}+\frac {\sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}{3 \sqrt [3]{2} x^2}}\right \},\left \{y(x)\to -\sqrt {\frac {2 i \sqrt [3]{2} x^2}{\sqrt {3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {2 \sqrt [3]{2} x^2}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {2 i \sqrt [3]{2} x}{\sqrt {3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {2 \sqrt [3]{2} x}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {i}{2^{2/3} \sqrt {3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {1}{3\ 2^{2/3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {2}{3}-\frac {2}{3 x}-\frac {i \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}{2 \sqrt [3]{2} \sqrt {3} x^2}-\frac {\sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}{6 \sqrt [3]{2} x^2}}\right \},\left \{y(x)\to \sqrt {\frac {2 i \sqrt [3]{2} x^2}{\sqrt {3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {2 \sqrt [3]{2} x^2}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {2 i \sqrt [3]{2} x}{\sqrt {3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {2 \sqrt [3]{2} x}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {i}{2^{2/3} \sqrt {3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {1}{3\ 2^{2/3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {2}{3}-\frac {2}{3 x}-\frac {i \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}{2 \sqrt [3]{2} \sqrt {3} x^2}-\frac {\sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}{6 \sqrt [3]{2} x^2}}\right \},\left \{y(x)\to -\sqrt {-\frac {2 i \sqrt [3]{2} x^2}{\sqrt {3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {2 \sqrt [3]{2} x^2}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {2 i \sqrt [3]{2} x}{\sqrt {3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {2 \sqrt [3]{2} x}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {i}{2^{2/3} \sqrt {3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {1}{3\ 2^{2/3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {2}{3}-\frac {2}{3 x}+\frac {i \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}{2 \sqrt [3]{2} \sqrt {3} x^2}-\frac {\sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}{6 \sqrt [3]{2} x^2}}\right \},\left \{y(x)\to \sqrt {-\frac {2 i \sqrt [3]{2} x^2}{\sqrt {3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {2 \sqrt [3]{2} x^2}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {2 i \sqrt [3]{2} x}{\sqrt {3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {2 \sqrt [3]{2} x}{3 \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {i}{2^{2/3} \sqrt {3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}-\frac {1}{3\ 2^{2/3} \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}+\frac {2}{3}-\frac {2}{3 x}+\frac {i \sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}{2 \sqrt [3]{2} \sqrt {3} x^2}-\frac {\sqrt [3]{16 x^6+24 x^5-27 c_1{}^2 x^4+12 x^4+2 x^3+3 \sqrt {3} \sqrt {-32 c_1{}^2 x^{10}-48 c_1{}^2 x^9+27 c_1{}^4 x^8-24 c_1{}^2 x^8-4 c_1{}^2 x^7}}}{6 \sqrt [3]{2} x^2}}\right \}\right \}$ Maple : cpu = 0.015 (sec), leaf count = 28

$\left \{ x+ \left ( y \left ( x \right ) \right ) ^{-2}-{\frac {{\it \_C1}}{ \left ( y \left ( x \right ) \right ) ^{2}}{\frac {1}{\sqrt { \left ( y \left ( x \right ) \right ) ^{2}-2}}}}=0,y \left ( x \right ) =0 \right \}$