#### 2.414   ODE No. 414

$x^3+x y'(x)^2+y(x) y'(x)=0$ Mathematica : cpu = 0.0749663 (sec), leaf count = 107

$\left \{\left \{y(x)\to x^2 \text {InverseFunction}\left [\int _1^{\text {\#1}}\frac {1}{5 K[1]+\sqrt {K[1]^2-4}}dK[1]\& \right ]\left [\int _1^x-\frac {1}{2 K[2]}dK[2]+c_1\right ]\right \},\left \{y(x)\to x^2 \text {InverseFunction}\left [\int _1^{\text {\#1}}\frac {1}{\sqrt {K[3]^2-4}-5 K[3]}dK[3]\& \right ]\left [\int _1^x\frac {1}{2 K[4]}dK[4]+c_1\right ]\right \}\right \}$ Maple : cpu = 0.273 (sec), leaf count = 269

$\left \{ \int _{{\it \_b}}^{x}\!{\frac {1}{{\it \_a}} \left ( -y \left ( x \right ) -\sqrt {-4\,{{\it \_a}}^{4}+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) \left ( \sqrt {-4\,{{\it \_a}}^{4}+ \left ( y \left ( x \right ) \right ) ^{2}}+5\,y \left ( x \right ) \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!{1 \left ( -2+ \left ( 80\,{\it \_f}+16\,\sqrt {-4\,{x}^{4}+{{\it \_f}}^{2}} \right ) \int _{{\it \_b}}^{x}\!{{{\it \_a}}^{3} \left ( \sqrt {-4\,{{\it \_a}}^{4}+{{\it \_f}}^{2}}+5\,{\it \_f} \right ) ^{-2}{\frac {1}{\sqrt {-4\,{{\it \_a}}^{4}+{{\it \_f}}^{2}}}}}\,{\rm d}{\it \_a} \right ) \left ( \sqrt {-4\,{x}^{4}+{{\it \_f}}^{2}}+5\,{\it \_f} \right ) ^{-1}}{d{\it \_f}}+{\it \_C1}=0,\int _{{\it \_b}}^{x}\!{\frac {1}{{\it \_a}} \left ( -y \left ( x \right ) +\sqrt {-4\,{{\it \_a}}^{4}+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) \left ( 5\,y \left ( x \right ) -\sqrt {-4\,{{\it \_a}}^{4}+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!{1 \left ( -2+ \left ( -80\,{\it \_f}+16\,\sqrt {-4\,{x}^{4}+{{\it \_f}}^{2}} \right ) \int _{{\it \_b}}^{x}\!{{{\it \_a}}^{3} \left ( -5\,{\it \_f}+\sqrt {-4\,{{\it \_a}}^{4}+{{\it \_f}}^{2}} \right ) ^{-2}{\frac {1}{\sqrt {-4\,{{\it \_a}}^{4}+{{\it \_f}}^{2}}}}}\,{\rm d}{\it \_a} \right ) \left ( 5\,{\it \_f}-\sqrt {-4\,{x}^{4}+{{\it \_f}}^{2}} \right ) ^{-1}}{d{\it \_f}}+{\it \_C1}=0 \right \}$