ODE No. 419

\[ x y'(x)^2+2 y(x) y'(x)-x=0 \] ✓ Mathematica : cpu = 0.949697 (sec), leaf count = 9073

DSolve[-x + 2*y[x]*Derivative[1][y][x] + x*Derivative[1][y][x]^2 == 0,y[x],x]
 

\[\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 = 0.045 (sec), leaf count = 109

dsolve(x*diff(y(x),x)^2+2*y(x)*diff(y(x),x)-x = 0,y(x))
 

\[x +\frac {\left (y \left (x \right )-\sqrt {y \left (x \right )^{2}+x^{2}}\right ) c_{1}}{\left (\frac {2 x^{2}+6 y \left (x \right )^{2}-6 y \left (x \right ) \sqrt {y \left (x \right )^{2}+x^{2}}}{x^{2}}\right )^{\frac {2}{3}} x} = 0\]