\[ x \left (x y'(x)-y(x)\right )^2-y'(x)=0 \] ✓ Mathematica : cpu = 2.36913 (sec), leaf count = 10121
DSolve[-Derivative[1][y][x] + x*(-y[x] + x*Derivative[1][y][x])^2 == 0,y[x],x]
\[\left \{\left \{y(x)\to -\frac {1}{2} \sqrt {-\frac {4}{9} \cosh (6 c_1) x^2+\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8+\cosh (18 c_1) x^2+\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}-\frac {1}{2} \sqrt {-\frac {8}{9} \cosh (6 c_1) x^2+\frac {8}{9} \sinh (6 c_1) x^2-\frac {-\frac {128}{27} \cosh (6 c_1)+\frac {128}{27} \sinh (6 c_1)-\frac {8}{729 x^6}}{4 \sqrt {-\frac {4}{9} \cosh (6 c_1) x^2+\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8+\cosh (18 c_1) x^2+\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8+\cosh (18 c_1) x^2+\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {2}{81 x^4}}-\frac {1}{18 x^2}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {-\frac {4}{9} \cosh (6 c_1) x^2+\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8+\cosh (18 c_1) x^2+\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}+\frac {1}{2} \sqrt {-\frac {8}{9} \cosh (6 c_1) x^2+\frac {8}{9} \sinh (6 c_1) x^2-\frac {-\frac {128}{27} \cosh (6 c_1)+\frac {128}{27} \sinh (6 c_1)-\frac {8}{729 x^6}}{4 \sqrt {-\frac {4}{9} \cosh (6 c_1) x^2+\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8+\cosh (18 c_1) x^2+\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8+\cosh (18 c_1) x^2+\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {2}{81 x^4}}-\frac {1}{18 x^2}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {-\frac {4}{9} \cosh (6 c_1) x^2+\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8+\cosh (18 c_1) x^2+\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}-\frac {1}{2} \sqrt {-\frac {8}{9} \cosh (6 c_1) x^2+\frac {8}{9} \sinh (6 c_1) x^2+\frac {-\frac {128}{27} \cosh (6 c_1)+\frac {128}{27} \sinh (6 c_1)-\frac {8}{729 x^6}}{4 \sqrt {-\frac {4}{9} \cosh (6 c_1) x^2+\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8+\cosh (18 c_1) x^2+\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8+\cosh (18 c_1) x^2+\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {2}{81 x^4}}-\frac {1}{18 x^2}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {-\frac {4}{9} \cosh (6 c_1) x^2+\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8+\cosh (18 c_1) x^2+\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}+\frac {1}{2} \sqrt {-\frac {8}{9} \cosh (6 c_1) x^2+\frac {8}{9} \sinh (6 c_1) x^2+\frac {-\frac {128}{27} \cosh (6 c_1)+\frac {128}{27} \sinh (6 c_1)-\frac {8}{729 x^6}}{4 \sqrt {-\frac {4}{9} \cosh (6 c_1) x^2+\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8+\cosh (18 c_1) x^2+\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8+\cosh (18 c_1) x^2+\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{-32 \cosh (18 c_1) x^{12}+32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6+\cosh (6 c_1)-\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {-4096 \cosh (42 c_1) x^{18}-4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}-48 \cosh (54 c_1) x^6-48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {2}{81 x^4}}-\frac {1}{18 x^2}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {\frac {4}{9} \cosh (6 c_1) x^2-\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (18 c_1) x^2-\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}-\frac {1}{2} \sqrt {\frac {8}{9} \cosh (6 c_1) x^2-\frac {8}{9} \sinh (6 c_1) x^2-\frac {\frac {128}{27} \cosh (6 c_1)-\frac {128}{27} \sinh (6 c_1)-\frac {8}{729 x^6}}{4 \sqrt {\frac {4}{9} \cosh (6 c_1) x^2-\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (18 c_1) x^2-\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (18 c_1) x^2-\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {2}{81 x^4}}-\frac {1}{18 x^2}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {\frac {4}{9} \cosh (6 c_1) x^2-\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (18 c_1) x^2-\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}+\frac {1}{2} \sqrt {\frac {8}{9} \cosh (6 c_1) x^2-\frac {8}{9} \sinh (6 c_1) x^2-\frac {\frac {128}{27} \cosh (6 c_1)-\frac {128}{27} \sinh (6 c_1)-\frac {8}{729 x^6}}{4 \sqrt {\frac {4}{9} \cosh (6 c_1) x^2-\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (18 c_1) x^2-\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (18 c_1) x^2-\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {2}{81 x^4}}-\frac {1}{18 x^2}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {\frac {4}{9} \cosh (6 c_1) x^2-\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (18 c_1) x^2-\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}-\frac {1}{2} \sqrt {\frac {8}{9} \cosh (6 c_1) x^2-\frac {8}{9} \sinh (6 c_1) x^2+\frac {\frac {128}{27} \cosh (6 c_1)-\frac {128}{27} \sinh (6 c_1)-\frac {8}{729 x^6}}{4 \sqrt {\frac {4}{9} \cosh (6 c_1) x^2-\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (18 c_1) x^2-\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (18 c_1) x^2-\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {2}{81 x^4}}-\frac {1}{18 x^2}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {\frac {4}{9} \cosh (6 c_1) x^2-\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (18 c_1) x^2-\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}+\frac {1}{2} \sqrt {\frac {8}{9} \cosh (6 c_1) x^2-\frac {8}{9} \sinh (6 c_1) x^2+\frac {\frac {128}{27} \cosh (6 c_1)-\frac {128}{27} \sinh (6 c_1)-\frac {8}{729 x^6}}{4 \sqrt {\frac {4}{9} \cosh (6 c_1) x^2-\frac {4}{9} \sinh (6 c_1) x^2+\frac {\sqrt [3]{2} \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}+\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (18 c_1) x^2-\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {1}{81 x^4}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}}}{9 x^2}-\frac {4\ 2^{2/3} \left (2 \cosh (12 c_1) x^8+2 \sinh (12 c_1) x^8-\cosh (18 c_1) x^2-\sinh (18 c_1) x^2\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{9 \sqrt [3]{32 \cosh (18 c_1) x^{12}-32 \sinh (18 c_1) x^{12}+40 \cosh (12 c_1) x^6-40 \sinh (12 c_1) x^6-\cosh (6 c_1)+\sinh (6 c_1)+(\cosh (36 c_1)-\sinh (36 c_1)) \sqrt {4096 \cosh (42 c_1) x^{18}+4096 \sinh (42 c_1) x^{18}+768 \cosh (48 c_1) x^{12}+768 \sinh (48 c_1) x^{12}+48 \cosh (54 c_1) x^6+48 \sinh (54 c_1) x^6+\cosh (60 c_1)+\sinh (60 c_1)}} x^2}+\frac {2}{81 x^4}}-\frac {1}{18 x^2}\right \}\right \}\] ✓ Maple : cpu = 1.949 (sec), leaf count = 221
dsolve(x*(x*diff(y(x),x)-y(x))^2-diff(y(x),x) = 0,y(x))
\[y \left (x \right ) = \frac {\left (\RootOf \left (-729 x^{12} c_{1}+\textit {\_Z}^{8}-12 \textit {\_Z}^{7}+60 \textit {\_Z}^{6}-160 \textit {\_Z}^{5}+240 \textit {\_Z}^{4}-192 \textit {\_Z}^{3}+64 \textit {\_Z}^{2}\right )-2\right ) \left (\RootOf \left (-729 x^{12} c_{1}+\textit {\_Z}^{8}-12 \textit {\_Z}^{7}+60 \textit {\_Z}^{6}-160 \textit {\_Z}^{5}+240 \textit {\_Z}^{4}-192 \textit {\_Z}^{3}+64 \textit {\_Z}^{2}\right )+1\right )}{9 x^{2}}\]