#### 2.419   ODE No. 419

$x y'(x)^2+2 y(x) y'(x)-x=0$ Mathematica : cpu = 1.05168 (sec), leaf count = 9073

$\left \{\left \{y(x)\to \frac {\cosh (3 c_1)}{18 x^2}+\frac {\sinh (3 c_1)}{18 x^2}-\frac {1}{2} \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}+\frac {\sinh (6 c_1)}{81 x^4}}-\frac {1}{2} \sqrt {\frac {8 x^2}{9}-\frac {-\frac {128}{27} \cosh (3 c_1)+\frac {8 \cosh (9 c_1)}{729 x^6}-\frac {128}{27} \sinh (3 c_1)+\frac {8 \sinh (9 c_1)}{729 x^6}}{4 \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}+\frac {\sinh (6 c_1)}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} \left (2 x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}} x^2}+\frac {2 \cosh (6 c_1)}{81 x^4}+\frac {2 \sinh (6 c_1)}{81 x^4}}\right \},\left \{y(x)\to \frac {\cosh (3 c_1)}{18 x^2}+\frac {\sinh (3 c_1)}{18 x^2}-\frac {1}{2} \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}+\frac {\sinh (6 c_1)}{81 x^4}}+\frac {1}{2} \sqrt {\frac {8 x^2}{9}-\frac {-\frac {128}{27} \cosh (3 c_1)+\frac {8 \cosh (9 c_1)}{729 x^6}-\frac {128}{27} \sinh (3 c_1)+\frac {8 \sinh (9 c_1)}{729 x^6}}{4 \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}+\frac {\sinh (6 c_1)}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} \left (2 x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}} x^2}+\frac {2 \cosh (6 c_1)}{81 x^4}+\frac {2 \sinh (6 c_1)}{81 x^4}}\right \},\left \{y(x)\to \frac {\cosh (3 c_1)}{18 x^2}+\frac {\sinh (3 c_1)}{18 x^2}+\frac {1}{2} \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}+\frac {\sinh (6 c_1)}{81 x^4}}-\frac {1}{2} \sqrt {\frac {8 x^2}{9}+\frac {-\frac {128}{27} \cosh (3 c_1)+\frac {8 \cosh (9 c_1)}{729 x^6}-\frac {128}{27} \sinh (3 c_1)+\frac {8 \sinh (9 c_1)}{729 x^6}}{4 \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}+\frac {\sinh (6 c_1)}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} \left (2 x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}} x^2}+\frac {2 \cosh (6 c_1)}{81 x^4}+\frac {2 \sinh (6 c_1)}{81 x^4}}\right \},\left \{y(x)\to \frac {\cosh (3 c_1)}{18 x^2}+\frac {\sinh (3 c_1)}{18 x^2}+\frac {1}{2} \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}+\frac {\sinh (6 c_1)}{81 x^4}}+\frac {1}{2} \sqrt {\frac {8 x^2}{9}+\frac {-\frac {128}{27} \cosh (3 c_1)+\frac {8 \cosh (9 c_1)}{729 x^6}-\frac {128}{27} \sinh (3 c_1)+\frac {8 \sinh (9 c_1)}{729 x^6}}{4 \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}+\frac {\sinh (6 c_1)}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} \left (2 x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6+40 \sinh (6 c_1) x^6-\cosh (12 c_1)-\sinh (12 c_1)+\sqrt {4096 \cosh (6 c_1) x^{18}+4096 \sinh (6 c_1) x^{18}+768 \cosh (12 c_1) x^{12}+768 \sinh (12 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (24 c_1)+\sinh (24 c_1)}} x^2}+\frac {2 \cosh (6 c_1)}{81 x^4}+\frac {2 \sinh (6 c_1)}{81 x^4}}\right \},\left \{y(x)\to -\frac {\cosh (3 c_1)}{18 x^2}+\frac {\sinh (3 c_1)}{18 x^2}-\frac {1}{2} \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} (\cosh (12 c_1)-\sinh (12 c_1)) \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}-\frac {\sinh (6 c_1)}{81 x^4}}-\frac {1}{2} \sqrt {\frac {8 x^2}{9}-\frac {\frac {128}{27} \cosh (3 c_1)-\frac {8 \cosh (9 c_1)}{729 x^6}-\frac {128}{27} \sinh (3 c_1)+\frac {8 \sinh (9 c_1)}{729 x^6}}{4 \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} (\cosh (12 c_1)-\sinh (12 c_1)) \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}-\frac {\sinh (6 c_1)}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} (\cosh (12 c_1)-\sinh (12 c_1)) \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}} x^2}+\frac {2 \cosh (6 c_1)}{81 x^4}-\frac {2 \sinh (6 c_1)}{81 x^4}}\right \},\left \{y(x)\to -\frac {\cosh (3 c_1)}{18 x^2}+\frac {\sinh (3 c_1)}{18 x^2}-\frac {1}{2} \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} (\cosh (12 c_1)-\sinh (12 c_1)) \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}-\frac {\sinh (6 c_1)}{81 x^4}}+\frac {1}{2} \sqrt {\frac {8 x^2}{9}-\frac {\frac {128}{27} \cosh (3 c_1)-\frac {8 \cosh (9 c_1)}{729 x^6}-\frac {128}{27} \sinh (3 c_1)+\frac {8 \sinh (9 c_1)}{729 x^6}}{4 \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} (\cosh (12 c_1)-\sinh (12 c_1)) \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}-\frac {\sinh (6 c_1)}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} (\cosh (12 c_1)-\sinh (12 c_1)) \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}} x^2}+\frac {2 \cosh (6 c_1)}{81 x^4}-\frac {2 \sinh (6 c_1)}{81 x^4}}\right \},\left \{y(x)\to -\frac {\cosh (3 c_1)}{18 x^2}+\frac {\sinh (3 c_1)}{18 x^2}+\frac {1}{2} \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} (\cosh (12 c_1)-\sinh (12 c_1)) \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}-\frac {\sinh (6 c_1)}{81 x^4}}-\frac {1}{2} \sqrt {\frac {8 x^2}{9}+\frac {\frac {128}{27} \cosh (3 c_1)-\frac {8 \cosh (9 c_1)}{729 x^6}-\frac {128}{27} \sinh (3 c_1)+\frac {8 \sinh (9 c_1)}{729 x^6}}{4 \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} (\cosh (12 c_1)-\sinh (12 c_1)) \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}-\frac {\sinh (6 c_1)}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} (\cosh (12 c_1)-\sinh (12 c_1)) \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}} x^2}+\frac {2 \cosh (6 c_1)}{81 x^4}-\frac {2 \sinh (6 c_1)}{81 x^4}}\right \},\left \{y(x)\to -\frac {\cosh (3 c_1)}{18 x^2}+\frac {\sinh (3 c_1)}{18 x^2}+\frac {1}{2} \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} (\cosh (12 c_1)-\sinh (12 c_1)) \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}-\frac {\sinh (6 c_1)}{81 x^4}}+\frac {1}{2} \sqrt {\frac {8 x^2}{9}+\frac {\frac {128}{27} \cosh (3 c_1)-\frac {8 \cosh (9 c_1)}{729 x^6}-\frac {128}{27} \sinh (3 c_1)+\frac {8 \sinh (9 c_1)}{729 x^6}}{4 \sqrt {\frac {4 x^2}{9}+\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} (\cosh (12 c_1)-\sinh (12 c_1)) \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}} x^2}+\frac {\cosh (6 c_1)}{81 x^4}-\frac {\sinh (6 c_1)}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} (\cosh (12 c_1)-\sinh (12 c_1)) \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (6 c_1) x^2-\sinh (6 c_1) x^2\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh (6 c_1) x^6-40 \sinh (6 c_1) x^6-\cosh (12 c_1)+\sinh (12 c_1)+(\cosh (18 c_1)-\sinh (18 c_1)) \sqrt {4096 \cosh (30 c_1) x^{18}+4096 \sinh (30 c_1) x^{18}+768 \cosh (24 c_1) x^{12}+768 \sinh (24 c_1) x^{12}+48 \cosh (18 c_1) x^6+48 \sinh (18 c_1) x^6+\cosh (12 c_1)+\sinh (12 c_1)}} x^2}+\frac {2 \cosh (6 c_1)}{81 x^4}-\frac {2 \sinh (6 c_1)}{81 x^4}}\right \}\right \}$ Maple : cpu = 0.059 (sec), leaf count = 109

$\left \{ x+{\frac {{\it \_C1}}{x} \left ( y \left ( x \right ) -\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}} \right ) \left ( {\frac {1}{{x}^{2}} \left ( 2\,{x}^{2}+6\, \left ( y \left ( x \right ) \right ) ^{2}-6\,y \left ( x \right ) \sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}} \right ) } \right ) ^{-{\frac {2}{3}}}}=0,{\frac {{\it \_C1}}{x} \left ( \sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}+y \left ( x \right ) \right ) \left ( {\frac {1}{{x}^{2}} \left ( 3\,y \left ( x \right ) \sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}+{x}^{2}+3\, \left ( y \left ( x \right ) \right ) ^{2} \right ) } \right ) ^{-{\frac {2}{3}}}}+x=0 \right \}$