4.19.34 \(x^3 y'(x)^2-\left (2 x^2 y(x)+1\right ) y'(x)+x y(x)^2=0\)

ODE
\[ x^3 y'(x)^2-\left (2 x^2 y(x)+1\right ) y'(x)+x y(x)^2=0 \] ODE Classification

[[_homogeneous, `class G`], _rational]

Book solution method
Change of variable

Mathematica ✓
cpu = 3.13579 (sec), leaf count = 8121

\[\left \{\left \{y(x)\to -\frac {\sqrt {-36 \cosh (6 c_1) x^2+36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6+\cosh (18 c_1)+\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}} x^2+\sqrt {-72 \cosh (6 c_1) x^2+72 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (\left (2 x^6+1\right ) \cosh (3 c_1)+\left (1-2 x^6\right ) \sinh (3 c_1)\right ) (\sinh (9 c_1)-\cosh (9 c_1))}{\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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}-\frac {2 \left (-432 \cosh (6 c_1)+432 \sinh (6 c_1)-\frac {1}{x^6}\right )}{\sqrt {-36 \cosh (6 c_1) x^2+36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6+\cosh (18 c_1)+\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}}}-\frac {9 \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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {2}{x^4}} x^2+1}{18 x^2}\right \},\left \{y(x)\to -\frac {\sqrt {-36 \cosh (6 c_1) x^2+36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6+\cosh (18 c_1)+\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}} x^2-\sqrt {-72 \cosh (6 c_1) x^2+72 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (\left (2 x^6+1\right ) \cosh (3 c_1)+\left (1-2 x^6\right ) \sinh (3 c_1)\right ) (\sinh (9 c_1)-\cosh (9 c_1))}{\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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}-\frac {2 \left (-432 \cosh (6 c_1)+432 \sinh (6 c_1)-\frac {1}{x^6}\right )}{\sqrt {-36 \cosh (6 c_1) x^2+36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6+\cosh (18 c_1)+\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}}}-\frac {9 \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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {2}{x^4}} x^2+1}{18 x^2}\right \},\left \{y(x)\to \frac {\sqrt {-36 \cosh (6 c_1) x^2+36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6+\cosh (18 c_1)+\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}} x^2-\sqrt {-72 \cosh (6 c_1) x^2+72 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (\left (2 x^6+1\right ) \cosh (3 c_1)+\left (1-2 x^6\right ) \sinh (3 c_1)\right ) (\sinh (9 c_1)-\cosh (9 c_1))}{\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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {2 \left (-432 \cosh (6 c_1)+432 \sinh (6 c_1)-\frac {1}{x^6}\right )}{\sqrt {-36 \cosh (6 c_1) x^2+36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6+\cosh (18 c_1)+\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}}}-\frac {9 \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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {2}{x^4}} x^2-1}{18 x^2}\right \},\left \{y(x)\to \frac {\sqrt {-36 \cosh (6 c_1) x^2+36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6+\cosh (18 c_1)+\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}} x^2+\sqrt {-72 \cosh (6 c_1) x^2+72 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (\left (2 x^6+1\right ) \cosh (3 c_1)+\left (1-2 x^6\right ) \sinh (3 c_1)\right ) (\sinh (9 c_1)-\cosh (9 c_1))}{\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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {2 \left (-432 \cosh (6 c_1)+432 \sinh (6 c_1)-\frac {1}{x^6}\right )}{\sqrt {-36 \cosh (6 c_1) x^2+36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6+\cosh (18 c_1)+\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}}}-\frac {9 \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 {-\left (\left (16 x^6-1\right ) \cosh (3 c_1)-\left (16 x^6+1\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {2}{x^4}} x^2-1}{18 x^2}\right \},\left \{y(x)\to -\frac {\sqrt {36 \cosh (6 c_1) x^2-36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6-\cosh (18 c_1)-\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}} x^2+\sqrt {72 \cosh (6 c_1) x^2-72 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (\left (2 x^6-1\right ) \cosh (3 c_1)-\left (2 x^6+1\right ) \sinh (3 c_1)\right ) (\sinh (9 c_1)-\cosh (9 c_1))}{\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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}-\frac {2 \left (432 \cosh (6 c_1)-432 \sinh (6 c_1)-\frac {1}{x^6}\right )}{\sqrt {36 \cosh (6 c_1) x^2-36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6-\cosh (18 c_1)-\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}}}-\frac {9 \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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {2}{x^4}} x^2+1}{18 x^2}\right \},\left \{y(x)\to -\frac {\sqrt {36 \cosh (6 c_1) x^2-36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6-\cosh (18 c_1)-\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}} x^2-\sqrt {72 \cosh (6 c_1) x^2-72 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (\left (2 x^6-1\right ) \cosh (3 c_1)-\left (2 x^6+1\right ) \sinh (3 c_1)\right ) (\sinh (9 c_1)-\cosh (9 c_1))}{\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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}-\frac {2 \left (432 \cosh (6 c_1)-432 \sinh (6 c_1)-\frac {1}{x^6}\right )}{\sqrt {36 \cosh (6 c_1) x^2-36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6-\cosh (18 c_1)-\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}}}-\frac {9 \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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {2}{x^4}} x^2+1}{18 x^2}\right \},\left \{y(x)\to \frac {\sqrt {36 \cosh (6 c_1) x^2-36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6-\cosh (18 c_1)-\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}} x^2-\sqrt {72 \cosh (6 c_1) x^2-72 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (\left (2 x^6-1\right ) \cosh (3 c_1)-\left (2 x^6+1\right ) \sinh (3 c_1)\right ) (\sinh (9 c_1)-\cosh (9 c_1))}{\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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {2 \left (432 \cosh (6 c_1)-432 \sinh (6 c_1)-\frac {1}{x^6}\right )}{\sqrt {36 \cosh (6 c_1) x^2-36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6-\cosh (18 c_1)-\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}}}-\frac {9 \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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {2}{x^4}} x^2-1}{18 x^2}\right \},\left \{y(x)\to \frac {\sqrt {36 \cosh (6 c_1) x^2-36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6-\cosh (18 c_1)-\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}} x^2+\sqrt {72 \cosh (6 c_1) x^2-72 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (\left (2 x^6-1\right ) \cosh (3 c_1)-\left (2 x^6+1\right ) \sinh (3 c_1)\right ) (\sinh (9 c_1)-\cosh (9 c_1))}{\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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {2 \left (432 \cosh (6 c_1)-432 \sinh (6 c_1)-\frac {1}{x^6}\right )}{\sqrt {36 \cosh (6 c_1) x^2-36 \sinh (6 c_1) x^2+\frac {36\ 2^{2/3} \left (2 \cosh (12 c_1) x^6+2 \sinh (12 c_1) x^6-\cosh (18 c_1)-\sinh (18 c_1)\right ) (\cosh (24 c_1)-\sinh (24 c_1))}{\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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}+\frac {9 \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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {1}{x^4}}}-\frac {9 \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 {\left (\left (16 x^6+1\right ) \cosh (3 c_1)+\left (1-16 x^6\right ) \sinh (3 c_1)\right ){}^3 (\cosh (51 c_1)+\sinh (51 c_1))}}}{x^2}+\frac {2}{x^4}} x^2-1}{18 x^2}\right \}\right \}\]

Maple ✓
cpu = 9.098 (sec), leaf count = 221

\[\left [y \left (x \right ) = \frac {\left (\RootOf \left (-729 \textit {\_C1} \,x^{12}+\textit {\_Z}^{8}+12 \textit {\_Z}^{7}+60 \textit {\_Z}^{6}+160 \textit {\_Z}^{5}+240 \textit {\_Z}^{4}+192 \textit {\_Z}^{3}+64 \textit {\_Z}^{2}\right )+2\right ) \left (-1+\RootOf \left (-729 \textit {\_C1} \,x^{12}+\textit {\_Z}^{8}+12 \textit {\_Z}^{7}+60 \textit {\_Z}^{6}+160 \textit {\_Z}^{5}+240 \textit {\_Z}^{4}+192 \textit {\_Z}^{3}+64 \textit {\_Z}^{2}\right )\right )}{9 x^{2}}, y \left (x \right ) = -\frac {2}{9 x^{2}}, y \left (x \right ) = \frac {\left (\RootOf \left (-729 x^{12}+\textit {\_C1} \,\textit {\_Z}^{8}-4 \textit {\_C1} \,\textit {\_Z}^{7}+4 \textit {\_C1} \,\textit {\_Z}^{6}+4 \textit {\_C1} \,\textit {\_Z}^{5}-10 \textit {\_C1} \,\textit {\_Z}^{4}+4 \textit {\_C1} \,\textit {\_Z}^{3}+4 \textit {\_Z}^{2} \textit {\_C1} -4 \textit {\_C1} \textit {\_Z} +\textit {\_C1} \right )+2\right ) \left (-1+\RootOf \left (-729 x^{12}+\textit {\_C1} \,\textit {\_Z}^{8}-4 \textit {\_C1} \,\textit {\_Z}^{7}+4 \textit {\_C1} \,\textit {\_Z}^{6}+4 \textit {\_C1} \,\textit {\_Z}^{5}-10 \textit {\_C1} \,\textit {\_Z}^{4}+4 \textit {\_C1} \,\textit {\_Z}^{3}+4 \textit {\_Z}^{2} \textit {\_C1} -4 \textit {\_C1} \textit {\_Z} +\textit {\_C1} \right )\right )}{9 x^{2}}\right ]\] Mathematica raw input

DSolve[x*y[x]^2 - (1 + 2*x^2*y[x])*y'[x] + x^3*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -1/18*(1 + x^2*Sqrt[x^(-4) - 36*x^2*Cosh[6*C[1]] + 36*x^2*Sinh[6*C[1]]
 + (36*2^(2/3)*(2*x^6*Cosh[12*C[1]] + Cosh[18*C[1]] + 2*x^6*Sinh[12*C[1]] + Sinh
[18*C[1]])*(Cosh[24*C[1]] - Sinh[24*C[1]]))/(Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]]
 - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18
*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1
 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3) + (9*2^(1/3)
*(Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 4
0*x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sq
rt[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] +
 Sinh[51*C[1]]))])^(1/3))/x^2] + x^2*Sqrt[2/x^4 - 72*x^2*Cosh[6*C[1]] + 72*x^2*S
inh[6*C[1]] + (36*2^(2/3)*((1 + 2*x^6)*Cosh[3*C[1]] + (1 - 2*x^6)*Sinh[3*C[1]])*
(-Cosh[9*C[1]] + Sinh[9*C[1]]))/(Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x^12*C
osh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18*C[1]] + (Co
sh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x^6)*S
inh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3) - (9*2^(1/3)*(Cosh[6*C[1
]] + 40*x^6*Cosh[12*C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*Sinh[1
2*C[1]] + 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((-1 + 
16*x^6)*Cosh[3*C[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1
]]))])^(1/3))/x^2 - (2*(-x^(-6) - 432*Cosh[6*C[1]] + 432*Sinh[6*C[1]]))/Sqrt[x^(
-4) - 36*x^2*Cosh[6*C[1]] + 36*x^2*Sinh[6*C[1]] + (36*2^(2/3)*(2*x^6*Cosh[12*C[1
]] + Cosh[18*C[1]] + 2*x^6*Sinh[12*C[1]] + Sinh[18*C[1]])*(Cosh[24*C[1]] - Sinh[
24*C[1]]))/(Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6
*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36
*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[
51*C[1]] + Sinh[51*C[1]]))])^(1/3) + (9*2^(1/3)*(Cosh[6*C[1]] + 40*x^6*Cosh[12*C
[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Sin
h[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]] 
- (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3))/x^2]])/
x^2}, {y[x] -> -1/18*(1 + x^2*Sqrt[x^(-4) - 36*x^2*Cosh[6*C[1]] + 36*x^2*Sinh[6*
C[1]] + (36*2^(2/3)*(2*x^6*Cosh[12*C[1]] + Cosh[18*C[1]] + 2*x^6*Sinh[12*C[1]] +
 Sinh[18*C[1]])*(Cosh[24*C[1]] - Sinh[24*C[1]]))/(Cosh[6*C[1]] + 40*x^6*Cosh[12*
C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Si
nh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]]
 - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3) + (9*2^
(1/3)*(Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]
] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]
])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[
1]] + Sinh[51*C[1]]))])^(1/3))/x^2] - x^2*Sqrt[2/x^4 - 72*x^2*Cosh[6*C[1]] + 72*
x^2*Sinh[6*C[1]] + (36*2^(2/3)*((1 + 2*x^6)*Cosh[3*C[1]] + (1 - 2*x^6)*Sinh[3*C[
1]])*(-Cosh[9*C[1]] + Sinh[9*C[1]]))/(Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x
^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18*C[1]] 
+ (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x
^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3) - (9*2^(1/3)*(Cosh[
6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*S
inh[12*C[1]] + 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((
-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[5
1*C[1]]))])^(1/3))/x^2 - (2*(-x^(-6) - 432*Cosh[6*C[1]] + 432*Sinh[6*C[1]]))/Sqr
t[x^(-4) - 36*x^2*Cosh[6*C[1]] + 36*x^2*Sinh[6*C[1]] + (36*2^(2/3)*(2*x^6*Cosh[1
2*C[1]] + Cosh[18*C[1]] + 2*x^6*Sinh[12*C[1]] + Sinh[18*C[1]])*(Cosh[24*C[1]] - 
Sinh[24*C[1]]))/(Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x^12*Cosh[18*C[1]] - S
inh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Si
nh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(
Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3) + (9*2^(1/3)*(Cosh[6*C[1]] + 40*x^6*Cosh
[12*C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^1
2*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C
[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3))/x^
2]])/x^2}, {y[x] -> (-1 + x^2*Sqrt[x^(-4) - 36*x^2*Cosh[6*C[1]] + 36*x^2*Sinh[6*
C[1]] + (36*2^(2/3)*(2*x^6*Cosh[12*C[1]] + Cosh[18*C[1]] + 2*x^6*Sinh[12*C[1]] +
 Sinh[18*C[1]])*(Cosh[24*C[1]] - Sinh[24*C[1]]))/(Cosh[6*C[1]] + 40*x^6*Cosh[12*
C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Si
nh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]]
 - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3) + (9*2^
(1/3)*(Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]
] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]
])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[
1]] + Sinh[51*C[1]]))])^(1/3))/x^2] - x^2*Sqrt[2/x^4 - 72*x^2*Cosh[6*C[1]] + 72*
x^2*Sinh[6*C[1]] + (36*2^(2/3)*((1 + 2*x^6)*Cosh[3*C[1]] + (1 - 2*x^6)*Sinh[3*C[
1]])*(-Cosh[9*C[1]] + Sinh[9*C[1]]))/(Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x
^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18*C[1]] 
+ (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x
^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3) - (9*2^(1/3)*(Cosh[
6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*S
inh[12*C[1]] + 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((
-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[5
1*C[1]]))])^(1/3))/x^2 + (2*(-x^(-6) - 432*Cosh[6*C[1]] + 432*Sinh[6*C[1]]))/Sqr
t[x^(-4) - 36*x^2*Cosh[6*C[1]] + 36*x^2*Sinh[6*C[1]] + (36*2^(2/3)*(2*x^6*Cosh[1
2*C[1]] + Cosh[18*C[1]] + 2*x^6*Sinh[12*C[1]] + Sinh[18*C[1]])*(Cosh[24*C[1]] - 
Sinh[24*C[1]]))/(Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x^12*Cosh[18*C[1]] - S
inh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Si
nh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(
Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3) + (9*2^(1/3)*(Cosh[6*C[1]] + 40*x^6*Cosh
[12*C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^1
2*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C
[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3))/x^
2]])/(18*x^2)}, {y[x] -> (-1 + x^2*Sqrt[x^(-4) - 36*x^2*Cosh[6*C[1]] + 36*x^2*Si
nh[6*C[1]] + (36*2^(2/3)*(2*x^6*Cosh[12*C[1]] + Cosh[18*C[1]] + 2*x^6*Sinh[12*C[
1]] + Sinh[18*C[1]])*(Cosh[24*C[1]] - Sinh[24*C[1]]))/(Cosh[6*C[1]] + 40*x^6*Cos
h[12*C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^
12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*
C[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3) + 
(9*2^(1/3)*(Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6
*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36
*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[
51*C[1]] + Sinh[51*C[1]]))])^(1/3))/x^2] + x^2*Sqrt[2/x^4 - 72*x^2*Cosh[6*C[1]] 
+ 72*x^2*Sinh[6*C[1]] + (36*2^(2/3)*((1 + 2*x^6)*Cosh[3*C[1]] + (1 - 2*x^6)*Sinh
[3*C[1]])*(-Cosh[9*C[1]] + Sinh[9*C[1]]))/(Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] -
 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18*C
[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1 +
 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3) - (9*2^(1/3)*(
Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*
x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt
[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + S
inh[51*C[1]]))])^(1/3))/x^2 + (2*(-x^(-6) - 432*Cosh[6*C[1]] + 432*Sinh[6*C[1]])
)/Sqrt[x^(-4) - 36*x^2*Cosh[6*C[1]] + 36*x^2*Sinh[6*C[1]] + (36*2^(2/3)*(2*x^6*C
osh[12*C[1]] + Cosh[18*C[1]] + 2*x^6*Sinh[12*C[1]] + Sinh[18*C[1]])*(Cosh[24*C[1
]] - Sinh[24*C[1]]))/(Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] - 32*x^12*Cosh[18*C[1]
] - Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]]
 - Sinh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cosh[3*C[1]] - (1 + 16*x^6)*Sinh[3*C[1]]
)^3*(Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3) + (9*2^(1/3)*(Cosh[6*C[1]] + 40*x^6
*Cosh[12*C[1]] - 32*x^12*Cosh[18*C[1]] - Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] + 3
2*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[-(((-1 + 16*x^6)*Cos
h[3*C[1]] - (1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]))])^(1/3
))/x^2]])/(18*x^2)}, {y[x] -> -1/18*(1 + x^2*Sqrt[x^(-4) + 36*x^2*Cosh[6*C[1]] -
 36*x^2*Sinh[6*C[1]] + (36*2^(2/3)*(2*x^6*Cosh[12*C[1]] - Cosh[18*C[1]] + 2*x^6*
Sinh[12*C[1]] - Sinh[18*C[1]])*(Cosh[24*C[1]] - Sinh[24*C[1]]))/(-Cosh[6*C[1]] +
 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[
1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)
*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(
1/3) + (9*2^(1/3)*(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]] 
+ Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] -
 Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(
Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3))/x^2] + x^2*Sqrt[2/x^4 + 72*x^2*Cosh[6*C[
1]] - 72*x^2*Sinh[6*C[1]] + (36*2^(2/3)*((-1 + 2*x^6)*Cosh[3*C[1]] - (1 + 2*x^6)
*Sinh[3*C[1]])*(-Cosh[9*C[1]] + Sinh[9*C[1]]))/(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C
[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sin
h[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (
1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3) - (9*2^(1/3)
*(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1]] - 
40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*S
qrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + S
inh[51*C[1]])])^(1/3))/x^2 - (2*(-x^(-6) + 432*Cosh[6*C[1]] - 432*Sinh[6*C[1]]))
/Sqrt[x^(-4) + 36*x^2*Cosh[6*C[1]] - 36*x^2*Sinh[6*C[1]] + (36*2^(2/3)*(2*x^6*Co
sh[12*C[1]] - Cosh[18*C[1]] + 2*x^6*Sinh[12*C[1]] - Sinh[18*C[1]])*(Cosh[24*C[1]
] - Sinh[24*C[1]]))/(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]
] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]]
 - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3
*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3) + (9*2^(1/3)*(-Cosh[6*C[1]] + 40*x^6*Co
sh[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x
^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[
1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3))/x^2]
])/x^2}, {y[x] -> -1/18*(1 + x^2*Sqrt[x^(-4) + 36*x^2*Cosh[6*C[1]] - 36*x^2*Sinh
[6*C[1]] + (36*2^(2/3)*(2*x^6*Cosh[12*C[1]] - Cosh[18*C[1]] + 2*x^6*Sinh[12*C[1]
] - Sinh[18*C[1]])*(Cosh[24*C[1]] - Sinh[24*C[1]]))/(-Cosh[6*C[1]] + 40*x^6*Cosh
[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^1
2*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]
] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3) + (9*2^
(1/3)*(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1
]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1
]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]
] + Sinh[51*C[1]])])^(1/3))/x^2] - x^2*Sqrt[2/x^4 + 72*x^2*Cosh[6*C[1]] - 72*x^2
*Sinh[6*C[1]] + (36*2^(2/3)*((-1 + 2*x^6)*Cosh[3*C[1]] - (1 + 2*x^6)*Sinh[3*C[1]
])*(-Cosh[9*C[1]] + Sinh[9*C[1]]))/(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] + 32*x^
12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*C[1]] +
 (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*
Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3) - (9*2^(1/3)*(-Cosh[6*C[
1]] + 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[
12*C[1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16
*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]
)])^(1/3))/x^2 - (2*(-x^(-6) + 432*Cosh[6*C[1]] - 432*Sinh[6*C[1]]))/Sqrt[x^(-4)
 + 36*x^2*Cosh[6*C[1]] - 36*x^2*Sinh[6*C[1]] + (36*2^(2/3)*(2*x^6*Cosh[12*C[1]] 
- Cosh[18*C[1]] + 2*x^6*Sinh[12*C[1]] - Sinh[18*C[1]])*(Cosh[24*C[1]] - Sinh[24*
C[1]]))/(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C
[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C
[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[
1]] + Sinh[51*C[1]])])^(1/3) + (9*2^(1/3)*(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] 
+ 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*
C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 1
6*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3))/x^2]])/x^2}, {y[
x] -> (-1 + x^2*Sqrt[x^(-4) + 36*x^2*Cosh[6*C[1]] - 36*x^2*Sinh[6*C[1]] + (36*2^
(2/3)*(2*x^6*Cosh[12*C[1]] - Cosh[18*C[1]] + 2*x^6*Sinh[12*C[1]] - Sinh[18*C[1]]
)*(Cosh[24*C[1]] - Sinh[24*C[1]]))/(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] + 32*x^
12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*C[1]] +
 (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*
Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3) + (9*2^(1/3)*(-Cosh[6*C[
1]] + 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[
12*C[1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16
*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]]
)])^(1/3))/x^2] - x^2*Sqrt[2/x^4 + 72*x^2*Cosh[6*C[1]] - 72*x^2*Sinh[6*C[1]] + (
36*2^(2/3)*((-1 + 2*x^6)*Cosh[3*C[1]] - (1 + 2*x^6)*Sinh[3*C[1]])*(-Cosh[9*C[1]]
 + Sinh[9*C[1]]))/(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]] 
+ Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] -
 Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(
Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3) - (9*2^(1/3)*(-Cosh[6*C[1]] + 40*x^6*Cosh
[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^1
2*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]
] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3))/x^2 + 
(2*(-x^(-6) + 432*Cosh[6*C[1]] - 432*Sinh[6*C[1]]))/Sqrt[x^(-4) + 36*x^2*Cosh[6*
C[1]] - 36*x^2*Sinh[6*C[1]] + (36*2^(2/3)*(2*x^6*Cosh[12*C[1]] - Cosh[18*C[1]] +
 2*x^6*Sinh[12*C[1]] - Sinh[18*C[1]])*(Cosh[24*C[1]] - Sinh[24*C[1]]))/(-Cosh[6*
C[1]] + 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sin
h[12*C[1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 
16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1
]])])^(1/3) + (9*2^(1/3)*(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18
*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*
C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1
]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3))/x^2]])/(18*x^2)}, {y[x] -> (-1 + 
x^2*Sqrt[x^(-4) + 36*x^2*Cosh[6*C[1]] - 36*x^2*Sinh[6*C[1]] + (36*2^(2/3)*(2*x^6
*Cosh[12*C[1]] - Cosh[18*C[1]] + 2*x^6*Sinh[12*C[1]] - Sinh[18*C[1]])*(Cosh[24*C
[1]] - Sinh[24*C[1]]))/(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C
[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[
1]] - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]]
)^3*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3) + (9*2^(1/3)*(-Cosh[6*C[1]] + 40*x^6
*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 3
2*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3
*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3))/x
^2] + x^2*Sqrt[2/x^4 + 72*x^2*Cosh[6*C[1]] - 72*x^2*Sinh[6*C[1]] + (36*2^(2/3)*(
(-1 + 2*x^6)*Cosh[3*C[1]] - (1 + 2*x^6)*Sinh[3*C[1]])*(-Cosh[9*C[1]] + Sinh[9*C[
1]]))/(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1
]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1
]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]
] + Sinh[51*C[1]])])^(1/3) - (9*2^(1/3)*(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] + 
32*x^12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*C[
1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*
x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3))/x^2 + (2*(-x^(-6) 
+ 432*Cosh[6*C[1]] - 432*Sinh[6*C[1]]))/Sqrt[x^(-4) + 36*x^2*Cosh[6*C[1]] - 36*x
^2*Sinh[6*C[1]] + (36*2^(2/3)*(2*x^6*Cosh[12*C[1]] - Cosh[18*C[1]] + 2*x^6*Sinh[
12*C[1]] - Sinh[18*C[1]])*(Cosh[24*C[1]] - Sinh[24*C[1]]))/(-Cosh[6*C[1]] + 40*x
^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sinh[6*C[1]] - 40*x^6*Sinh[12*C[1]] -
 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh
[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[51*C[1]] + Sinh[51*C[1]])])^(1/3) 
+ (9*2^(1/3)*(-Cosh[6*C[1]] + 40*x^6*Cosh[12*C[1]] + 32*x^12*Cosh[18*C[1]] + Sin
h[6*C[1]] - 40*x^6*Sinh[12*C[1]] - 32*x^12*Sinh[18*C[1]] + (Cosh[36*C[1]] - Sinh
[36*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[
51*C[1]] + Sinh[51*C[1]])])^(1/3))/x^2]])/(18*x^2)}}

Maple raw input

dsolve(x^3*diff(y(x),x)^2-(1+2*x^2*y(x))*diff(y(x),x)+x*y(x)^2 = 0, y(x))

Maple raw output

[y(x) = 1/9*(RootOf(-729*_C1*x^12+_Z^8+12*_Z^7+60*_Z^6+160*_Z^5+240*_Z^4+192*_Z^
3+64*_Z^2)+2)*(-1+RootOf(-729*_C1*x^12+_Z^8+12*_Z^7+60*_Z^6+160*_Z^5+240*_Z^4+19
2*_Z^3+64*_Z^2))/x^2, y(x) = -2/9/x^2, y(x) = 1/9*(RootOf(-729*x^12+_C1*_Z^8-4*_
C1*_Z^7+4*_C1*_Z^6+4*_C1*_Z^5-10*_C1*_Z^4+4*_C1*_Z^3+4*_C1*_Z^2-4*_C1*_Z+_C1)+2)
*(-1+RootOf(-729*x^12+_C1*_Z^8-4*_C1*_Z^7+4*_C1*_Z^6+4*_C1*_Z^5-10*_C1*_Z^4+4*_C
1*_Z^3+4*_C1*_Z^2-4*_C1*_Z+_C1))/x^2]