#### 2.311   ODE No. 311

$4 x^3+9 x^2 y(x)+\left (3 x^3+6 x^2 y(x)-3 x y(x)^2+20 y(x)^3\right ) y'(x)+6 x y(x)^2-y(x)^3=0$ Mathematica : cpu = 0.22382 (sec), leaf count = 2201

$\left \{\left \{y(x)\to \frac {x}{20}+\frac {1}{2} \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}-\frac {1}{2} \sqrt {-\frac {659 x^3}{500 \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}}-\frac {39 x^2}{50}-\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}-\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}\right \},\left \{y(x)\to \frac {x}{20}+\frac {1}{2} \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}+\frac {1}{2} \sqrt {-\frac {659 x^3}{500 \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}}-\frac {39 x^2}{50}-\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}-\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}\right \},\left \{y(x)\to \frac {x}{20}-\frac {1}{2} \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}-\frac {1}{2} \sqrt {\frac {659 x^3}{500 \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}}-\frac {39 x^2}{50}-\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}-\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}\right \},\left \{y(x)\to \frac {x}{20}-\frac {1}{2} \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}+\frac {1}{2} \sqrt {\frac {659 x^3}{500 \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}}-\frac {39 x^2}{50}-\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}-\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}\right \}\right \}$ Maple : cpu = 0.067 (sec), leaf count = 50

$\left \{ y \left ( x \right ) ={\frac {{\it RootOf} \left ( {x}^{4}{{\it \_C1}}^{4}+3\,{x}^{3}{{\it \_C1}}^{3}{\it \_Z}+3\,{{\it \_C1}}^{2}{{\it \_Z}}^{2}{x}^{2}-{\it \_C1}\,{{\it \_Z}}^{3}x+5\,{{\it \_Z}}^{4}-1 \right ) }{{\it \_C1}}} \right \}$