#### 2.529   ODE No. 529

$y'(x)^3+x y'(x)^2-y(x)=0$ Mathematica : cpu = 39.3703 (sec), leaf count = 1758

$\left \{\left \{y(x)\to \frac {1}{2} \left (\frac {4\ 2^{2/3} x^4}{3 \left (-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54\right ){}^{2/3}}+\frac {4 \sqrt [3]{2} x^3}{3 \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}+\frac {8\ 2^{2/3} x^3}{\left (-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54\right ){}^{2/3}}+\frac {6 \sqrt [3]{2} x^2}{\sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}+\frac {18\ 2^{2/3} x^2}{\left (-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54\right ){}^{2/3}}-\frac {x^2}{3}+\frac {1}{3} 2^{2/3} \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54} x-\frac {\sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54} x}{3 \sqrt [3]{2}}+\frac {9 \sqrt [3]{2} x}{\sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}+\frac {18\ 2^{2/3} x}{\left (-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54\right ){}^{2/3}}+x+2 c_1+\frac {\left (-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54\right ){}^{2/3}}{12\ 2^{2/3}}+\frac {\sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}{2 \sqrt [3]{2}}+\frac {9}{2^{2/3} \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}+\frac {27}{2 \sqrt [3]{2} \left (-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54\right ){}^{2/3}}+\frac {9}{4}\right )\right \},\left \{y(x)\to \frac {1}{2} \left (3 \left (\frac {1}{6} (3-2 x)-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}{12 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (-4 x^2-12 x-9\right )}{6\ 2^{2/3} \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}\right ){}^2+4 x \left (\frac {1}{6} (3-2 x)-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}{12 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (-4 x^2-12 x-9\right )}{6\ 2^{2/3} \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}\right )-2 x+2 c_1\right )\right \},\left \{y(x)\to \frac {1}{2} \left (3 \left (\frac {1}{6} (3-2 x)-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}{12 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (-4 x^2-12 x-9\right )}{6\ 2^{2/3} \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}\right ){}^2+4 x \left (\frac {1}{6} (3-2 x)-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}{12 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (-4 x^2-12 x-9\right )}{6\ 2^{2/3} \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}\right )-2 x+2 c_1\right )\right \}\right \}$ Maple : cpu = 0.037 (sec), leaf count = 1251

$\left \{ y \left ( x \right ) =0,y \left ( x \right ) ={1 \left ( \left ( 4\,x-6 \right ) \sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) }}+ \left ( i \left ( -36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) } \right ) ^{{\frac {2}{3}}}-4\,i{x}^{2}-12\,ix-9\,i \right ) \sqrt {3}+ \left ( -36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) } \right ) ^{{\frac {2}{3}}}+4\,{x}^{2}+12\,x+9 \right ) ^{2} \left ( \left ( -8\,x-6 \right ) \sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) }}+ \left ( i \left ( -36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) } \right ) ^{{\frac {2}{3}}}-4\,i{x}^{2}-12\,ix-9\,i \right ) \sqrt {3}+ \left ( -36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) } \right ) ^{{\frac {2}{3}}}+4\,{x}^{2}+12\,x+9 \right ) \left ( 13824\,{x}^{3}+62208\,{x}^{2}+93312\,x-186624\,{\it \_C1}-10368\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) }-46656 \right ) ^{-1}},y \left ( x \right ) =-{1 \left ( \left ( -4\,x+6 \right ) \sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) }}+ \left ( i \left ( -36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) } \right ) ^{{\frac {2}{3}}}-4\,i{x}^{2}-12\,ix-9\,i \right ) \sqrt {3}- \left ( -36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) } \right ) ^{{\frac {2}{3}}}-4\,{x}^{2}-12\,x-9 \right ) ^{2} \left ( \left ( 8\,x+6 \right ) \sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) }}+ \left ( i \left ( -36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) } \right ) ^{{\frac {2}{3}}}-4\,i{x}^{2}-12\,ix-9\,i \right ) \sqrt {3}- \left ( -36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) } \right ) ^{{\frac {2}{3}}}-4\,{x}^{2}-12\,x-9 \right ) \left ( 13824\,{x}^{3}+62208\,{x}^{2}+93312\,x-186624\,{\it \_C1}-10368\,\sqrt {-6\, \left ( 1+2\,{\it \_C1} \right ) \left ( 4\,{x}^{3}+18\,{x}^{2}-27\,{\it \_C1}+27\,x \right ) }-46656 \right ) ^{-1}},y \left ( x \right ) ={\frac {1}{216} \left ( 4\,{\frac {{x}^{2}}{\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,x{\it \_C1}-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}}+12\,{\frac {x}{\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,x{\it \_C1}-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}}-2\,x+9\,{\frac {1}{\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,x{\it \_C1}-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}}+\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,x{\it \_C1}-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}+3 \right ) ^{3}}+{\frac {4\,x}{9} \left ( {1 \left ( x+{\frac {3}{2}} \right ) ^{2}{\frac {1}{\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,x{\it \_C1}-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}}}-{\frac {x}{2}}+{\frac {1}{4}\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,x{\it \_C1}-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}+{\frac {3}{4}} \right ) ^{2}} \right \}$