#### 2.380   ODE No. 380

$y'(x)^2+2 x y'(x)-y(x)=0$ Mathematica : cpu = 0.378949 (sec), leaf count = 1757

$\left \{\left \{y(x)\to -\frac {x^2}{4}-\frac {1}{4} \sqrt [3]{x^6-20 \cosh (3 c_1) x^3-20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+8 \sqrt {-\cosh (3 c_1) x^9-\sinh (3 c_1) x^9+3 \cosh (6 c_1) x^6+3 \sinh (6 c_1) x^6-3 \cosh (9 c_1) x^3-3 \sinh (9 c_1) x^3+\cosh (12 c_1)+\sinh (12 c_1)}}+\frac {-9 x^4-72 \cosh (3 c_1) x-72 \sinh (3 c_1) x}{36 \sqrt [3]{x^6-20 \cosh (3 c_1) x^3-20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+8 \sqrt {-\cosh (3 c_1) x^9-\sinh (3 c_1) x^9+3 \cosh (6 c_1) x^6+3 \sinh (6 c_1) x^6-3 \cosh (9 c_1) x^3-3 \sinh (9 c_1) x^3+\cosh (12 c_1)+\sinh (12 c_1)}}}\right \},\left \{y(x)\to -\frac {x^2}{4}+\frac {1}{8} \left (1-i \sqrt {3}\right ) \sqrt [3]{x^6-20 \cosh (3 c_1) x^3-20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+8 \sqrt {-\cosh (3 c_1) x^9-\sinh (3 c_1) x^9+3 \cosh (6 c_1) x^6+3 \sinh (6 c_1) x^6-3 \cosh (9 c_1) x^3-3 \sinh (9 c_1) x^3+\cosh (12 c_1)+\sinh (12 c_1)}}-\frac {\left (1+i \sqrt {3}\right ) \left (-9 x^4-72 \cosh (3 c_1) x-72 \sinh (3 c_1) x\right )}{72 \sqrt [3]{x^6-20 \cosh (3 c_1) x^3-20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+8 \sqrt {-\cosh (3 c_1) x^9-\sinh (3 c_1) x^9+3 \cosh (6 c_1) x^6+3 \sinh (6 c_1) x^6-3 \cosh (9 c_1) x^3-3 \sinh (9 c_1) x^3+\cosh (12 c_1)+\sinh (12 c_1)}}}\right \},\left \{y(x)\to -\frac {x^2}{4}+\frac {1}{8} \left (1+i \sqrt {3}\right ) \sqrt [3]{x^6-20 \cosh (3 c_1) x^3-20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+8 \sqrt {-\cosh (3 c_1) x^9-\sinh (3 c_1) x^9+3 \cosh (6 c_1) x^6+3 \sinh (6 c_1) x^6-3 \cosh (9 c_1) x^3-3 \sinh (9 c_1) x^3+\cosh (12 c_1)+\sinh (12 c_1)}}-\frac {\left (1-i \sqrt {3}\right ) \left (-9 x^4-72 \cosh (3 c_1) x-72 \sinh (3 c_1) x\right )}{72 \sqrt [3]{x^6-20 \cosh (3 c_1) x^3-20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+8 \sqrt {-\cosh (3 c_1) x^9-\sinh (3 c_1) x^9+3 \cosh (6 c_1) x^6+3 \sinh (6 c_1) x^6-3 \cosh (9 c_1) x^3-3 \sinh (9 c_1) x^3+\cosh (12 c_1)+\sinh (12 c_1)}}}\right \},\left \{y(x)\to -\frac {x^2}{4}-\frac {1}{4} \sqrt [3]{x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+8 \sqrt {\cosh (3 c_1) x^9+\sinh (3 c_1) x^9+3 \cosh (6 c_1) x^6+3 \sinh (6 c_1) x^6+3 \cosh (9 c_1) x^3+3 \sinh (9 c_1) x^3+\cosh (12 c_1)+\sinh (12 c_1)}}+\frac {-9 x^4+72 \cosh (3 c_1) x+72 \sinh (3 c_1) x}{36 \sqrt [3]{x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+8 \sqrt {\cosh (3 c_1) x^9+\sinh (3 c_1) x^9+3 \cosh (6 c_1) x^6+3 \sinh (6 c_1) x^6+3 \cosh (9 c_1) x^3+3 \sinh (9 c_1) x^3+\cosh (12 c_1)+\sinh (12 c_1)}}}\right \},\left \{y(x)\to -\frac {x^2}{4}+\frac {1}{8} \left (1-i \sqrt {3}\right ) \sqrt [3]{x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+8 \sqrt {\cosh (3 c_1) x^9+\sinh (3 c_1) x^9+3 \cosh (6 c_1) x^6+3 \sinh (6 c_1) x^6+3 \cosh (9 c_1) x^3+3 \sinh (9 c_1) x^3+\cosh (12 c_1)+\sinh (12 c_1)}}-\frac {\left (1+i \sqrt {3}\right ) \left (-9 x^4+72 \cosh (3 c_1) x+72 \sinh (3 c_1) x\right )}{72 \sqrt [3]{x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+8 \sqrt {\cosh (3 c_1) x^9+\sinh (3 c_1) x^9+3 \cosh (6 c_1) x^6+3 \sinh (6 c_1) x^6+3 \cosh (9 c_1) x^3+3 \sinh (9 c_1) x^3+\cosh (12 c_1)+\sinh (12 c_1)}}}\right \},\left \{y(x)\to -\frac {x^2}{4}+\frac {1}{8} \left (1+i \sqrt {3}\right ) \sqrt [3]{x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+8 \sqrt {\cosh (3 c_1) x^9+\sinh (3 c_1) x^9+3 \cosh (6 c_1) x^6+3 \sinh (6 c_1) x^6+3 \cosh (9 c_1) x^3+3 \sinh (9 c_1) x^3+\cosh (12 c_1)+\sinh (12 c_1)}}-\frac {\left (1-i \sqrt {3}\right ) \left (-9 x^4+72 \cosh (3 c_1) x+72 \sinh (3 c_1) x\right )}{72 \sqrt [3]{x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+8 \sqrt {\cosh (3 c_1) x^9+\sinh (3 c_1) x^9+3 \cosh (6 c_1) x^6+3 \sinh (6 c_1) x^6+3 \cosh (9 c_1) x^3+3 \sinh (9 c_1) x^3+\cosh (12 c_1)+\sinh (12 c_1)}}}\right \}\right \}$ Maple : cpu = 0.032 (sec), leaf count = 619

$\left \{ y \left ( x \right ) ={\frac {1}{16} \left ( i \left ( 6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}\sqrt {3}-i\sqrt {3}{x}^{2}- \left ( 6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}-2\,x\sqrt [3]{6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}}}-{x}^{2} \right ) \left ( i \left ( 6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}\sqrt {3}-i\sqrt {3}{x}^{2}- \left ( 6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}+6\,x\sqrt [3]{6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}}}-{x}^{2} \right ) \left ( 6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{-{\frac {2}{3}}}},y \left ( x \right ) ={\frac {1}{16} \left ( i \left ( 6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}\sqrt {3}-i\sqrt {3}{x}^{2}+ \left ( 6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}+2\,x\sqrt [3]{6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}}}+{x}^{2} \right ) \left ( i \left ( 6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}\sqrt {3}-i\sqrt {3}{x}^{2}+ \left ( 6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}-6\,x\sqrt [3]{6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}}}+{x}^{2} \right ) \left ( 6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{-{\frac {2}{3}}}},y \left ( x \right ) ={\frac {1}{4} \left ( {{x}^{2}{\frac {1}{\sqrt [3]{6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}}}}}}-x+\sqrt [3]{6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}}} \right ) ^{2}}+x \left ( {{x}^{2}{\frac {1}{\sqrt [3]{6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}}}}}}-x+\sqrt [3]{6\,{\it \_C1}-{x}^{3}+2\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+9\,{{\it \_C1}}^{2}}} \right ) \right \}$