ODE
\[ x y'(x)^2+2 y(x) y'(x)-x=0 \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
Homogeneous ODE, \(x^n f\left ( \frac {y}{x} , y' \right )=0\), Solve for \(p\)
Mathematica ✓
cpu = 1.46566 (sec), leaf count = 6977
\[\left \{\left \{y(x)\to \frac {-\sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (2 x^6-\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )}{x^4}} x^2-9 \sqrt {\frac {8 x^2}{9}-\frac {4\ 2^{2/3} \left (2 x^6-\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}{9 x^2}+\frac {2 \cosh \left (6 c_1\right )}{81 x^4}+\frac {2 \sinh \left (6 c_1\right )}{81 x^4}-\frac {2 \left (-432 \cosh \left (3 c_1\right ) x^6-432 \sinh \left (3 c_1\right ) x^6+\cosh \left (9 c_1\right )+\sinh \left (9 c_1\right )\right )}{81 \sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (2 x^6-\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )}{x^4}} x^6}} x^2+\cosh \left (3 c_1\right )+\sinh \left (3 c_1\right )}{18 x^2}\right \},\left \{y(x)\to \frac {-\sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (2 x^6-\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )}{x^4}} x^2+9 \sqrt {\frac {8 x^2}{9}-\frac {4\ 2^{2/3} \left (2 x^6-\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}{9 x^2}+\frac {2 \cosh \left (6 c_1\right )}{81 x^4}+\frac {2 \sinh \left (6 c_1\right )}{81 x^4}-\frac {2 \left (-432 \cosh \left (3 c_1\right ) x^6-432 \sinh \left (3 c_1\right ) x^6+\cosh \left (9 c_1\right )+\sinh \left (9 c_1\right )\right )}{81 \sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (2 x^6-\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )}{x^4}} x^6}} x^2+\cosh \left (3 c_1\right )+\sinh \left (3 c_1\right )}{18 x^2}\right \},\left \{y(x)\to \frac {\sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (2 x^6-\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )}{x^4}} x^2-9 \sqrt {\frac {8 x^2}{9}-\frac {4\ 2^{2/3} \left (2 x^6-\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}{9 x^2}+\frac {2 \cosh \left (6 c_1\right )}{81 x^4}+\frac {2 \sinh \left (6 c_1\right )}{81 x^4}+\frac {2 \left (-432 \cosh \left (3 c_1\right ) x^6-432 \sinh \left (3 c_1\right ) x^6+\cosh \left (9 c_1\right )+\sinh \left (9 c_1\right )\right )}{81 \sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (2 x^6-\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )}{x^4}} x^6}} x^2+\cosh \left (3 c_1\right )+\sinh \left (3 c_1\right )}{18 x^2}\right \},\left \{y(x)\to \frac {\sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (2 x^6-\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )}{x^4}} x^2+9 \sqrt {\frac {8 x^2}{9}-\frac {4\ 2^{2/3} \left (2 x^6-\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )\right )}{9 \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}-\frac {\sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}{9 x^2}+\frac {2 \cosh \left (6 c_1\right )}{81 x^4}+\frac {2 \sinh \left (6 c_1\right )}{81 x^4}+\frac {2 \left (-432 \cosh \left (3 c_1\right ) x^6-432 \sinh \left (3 c_1\right ) x^6+\cosh \left (9 c_1\right )+\sinh \left (9 c_1\right )\right )}{81 \sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (2 x^6-\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6+40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )-\sinh \left (12 c_1\right )+\sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (1-16 x^6\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (15 c_1\right )+\sinh \left (15 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )}{x^4}} x^6}} x^2+\cosh \left (3 c_1\right )+\sinh \left (3 c_1\right )}{18 x^2}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )}{x^4}} x^2+\sqrt {72 x^2+\frac {36\ 2^{2/3} \left (\sinh \left (3 c_1\right )-\cosh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}-\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^2}+\frac {2 \cosh \left (6 c_1\right )}{x^4}-\frac {2 \sinh \left (6 c_1\right )}{x^4}-\frac {2 \left (432 \cosh \left (3 c_1\right ) x^6-432 \sinh \left (3 c_1\right ) x^6-\cosh \left (9 c_1\right )+\sinh \left (9 c_1\right )\right )}{\sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )}{x^4}} x^6}} x^2+\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )}{18 x^2}\right \},\left \{y(x)\to \frac {-\sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )}{x^4}} x^2+\sqrt {72 x^2+\frac {36\ 2^{2/3} \left (\sinh \left (3 c_1\right )-\cosh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}-\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^2}+\frac {2 \cosh \left (6 c_1\right )}{x^4}-\frac {2 \sinh \left (6 c_1\right )}{x^4}-\frac {2 \left (432 \cosh \left (3 c_1\right ) x^6-432 \sinh \left (3 c_1\right ) x^6-\cosh \left (9 c_1\right )+\sinh \left (9 c_1\right )\right )}{\sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )}{x^4}} x^6}} x^2-\cosh \left (3 c_1\right )+\sinh \left (3 c_1\right )}{18 x^2}\right \},\left \{y(x)\to \frac {\sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )}{x^4}} x^2-\sqrt {72 x^2+\frac {36\ 2^{2/3} \left (\sinh \left (3 c_1\right )-\cosh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}-\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^2}+\frac {2 \cosh \left (6 c_1\right )}{x^4}-\frac {2 \sinh \left (6 c_1\right )}{x^4}+\frac {2 \left (432 \cosh \left (3 c_1\right ) x^6-432 \sinh \left (3 c_1\right ) x^6-\cosh \left (9 c_1\right )+\sinh \left (9 c_1\right )\right )}{\sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )}{x^4}} x^6}} x^2-\cosh \left (3 c_1\right )+\sinh \left (3 c_1\right )}{18 x^2}\right \},\left \{y(x)\to \frac {\sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )}{x^4}} x^2+\sqrt {72 x^2+\frac {36\ 2^{2/3} \left (\sinh \left (3 c_1\right )-\cosh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}-\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^2}+\frac {2 \cosh \left (6 c_1\right )}{x^4}-\frac {2 \sinh \left (6 c_1\right )}{x^4}+\frac {2 \left (432 \cosh \left (3 c_1\right ) x^6-432 \sinh \left (3 c_1\right ) x^6-\cosh \left (9 c_1\right )+\sinh \left (9 c_1\right )\right )}{\sqrt {\frac {36 x^6+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right ) x^4}{\sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+9 \sqrt [3]{2} \sqrt [3]{32 x^{12}+40 \cosh \left (6 c_1\right ) x^6-40 \sinh \left (6 c_1\right ) x^6-\cosh \left (12 c_1\right )+\sinh \left (12 c_1\right )+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {\left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}} x^2+\cosh \left (6 c_1\right )-\sinh \left (6 c_1\right )}{x^4}} x^6}} x^2-\cosh \left (3 c_1\right )+\sinh \left (3 c_1\right )}{18 x^2}\right \}\right \}\]
Maple ✓
cpu = 0.029 (sec), leaf count = 37
\[ \left \{ [x \left ( {\it \_T} \right ) ={{\it \_T}\,{\it \_C1} \left ( 3\,{{\it \_T}}^{2}-1 \right ) ^{-{\frac {2}{3}}}},y \left ( {\it \_T} \right ) =-{\frac { \left ( {{\it \_T}}^{2}-1 \right ) {\it \_C1}}{2} \left ( 3\,{{\it \_T}}^{2}-1 \right ) ^{-{\frac {2}{3}}}}] \right \} \] Mathematica raw input
DSolve[-x + 2*y[x]*y'[x] + x*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> (Cosh[3*C[1]] + Sinh[3*C[1]] - x^2*Sqrt[(36*x^6 + Cosh[6*C[1]] + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/x^4] - 9*x^2*Sqrt[(8*x^2)/9 + (2*Cosh[6*C[1]])/(81*x^4) + (2*Sinh[6*C[1]])/(81*x^4) - (4*2^(2/3)*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(9*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3)) - (2^(1/3)*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/(9*x^2) - (2*(-432*x^6*Cosh[3*C[1]] + Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(81*x^6*Sqrt[(36*x^6 + Cosh[6*C[1]] + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/x^4])])/(18*x^2)}, {y[x] -> (Cosh[3*C[1]] + Sinh[3*C[1]] - x^2*Sqrt[(36*x^6 + Cosh[6*C[1]] + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/x^4] + 9*x^2*Sqrt[(8*x^2)/9 + (2*Cosh[6*C[1]])/(81*x^4) + (2*Sinh[6*C[1]])/(81*x^4) - (4*2^(2/3)*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(9*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3)) - (2^(1/3)*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/(9*x^2) - (2*(-432*x^6*Cosh[3*C[1]] + Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(81*x^6*Sqrt[(36*x^6 + Cosh[6*C[1]] + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/x^4])])/(18*x^2)}, {y[x] -> (Cosh[3*C[1]] + Sinh[3*C[1]] + x^2*Sqrt[(36*x^6 + Cosh[6*C[1]] + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/x^4] - 9*x^2*Sqrt[(8*x^2)/9 + (2*Cosh[6*C[1]])/(81*x^4) + (2*Sinh[6*C[1]])/(81*x^4) - (4*2^(2/3)*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(9*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3)) - (2^(1/3)*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/(9*x^2) + (2*(-432*x^6*Cosh[3*C[1]] + Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(81*x^6*Sqrt[(36*x^6 + Cosh[6*C[1]] + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/x^4])])/(18*x^2)}, {y[x] -> (Cosh[3*C[1]] + Sinh[3*C[1]] + x^2*Sqrt[(36*x^6 + Cosh[6*C[1]] + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/x^4] + 9*x^2*Sqrt[(8*x^2)/9 + (2*Cosh[6*C[1]])/(81*x^4) + (2*Sinh[6*C[1]])/(81*x^4) - (4*2^(2/3)*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(9*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3)) - (2^(1/3)*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/(9*x^2) + (2*(-432*x^6*Cosh[3*C[1]] + Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(81*x^6*Sqrt[(36*x^6 + Cosh[6*C[1]] + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/x^4])])/(18*x^2)}, {y[x] -> -(Cosh[3*C[1]] - Sinh[3*C[1]] + x^2*Sqrt[(36*x^6 + Cosh[6*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^4] + x^2*Sqrt[72*x^2 + (2*Cosh[6*C[1]])/x^4 - (2*Sinh[6*C[1]])/x^4 + (36*2^(2/3)*(-Cosh[3*C[1]] + Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) - (9*2^(1/3)*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^2 - (2*(432*x^6*Cosh[3*C[1]] - Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(x^6*Sqrt[(36*x^6 + Cosh[6*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^4])])/(18*x^2)}, {y[x] -> (-Cosh[3*C[1]] + Sinh[3*C[1]] - x^2*Sqrt[(36*x^6 + Cosh[6*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^4] + x^2*Sqrt[72*x^2 + (2*Cosh[6*C[1]])/x^4 - (2*Sinh[6*C[1]])/x^4 + (36*2^(2/3)*(-Cosh[3*C[1]] + Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) - (9*2^(1/3)*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^2 - (2*(432*x^6*Cosh[3*C[1]] - Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(x^6*Sqrt[(36*x^6 + Cosh[6*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^4])])/(18*x^2)}, {y[x] -> (-Cosh[3*C[1]] + Sinh[3*C[1]] + x^2*Sqrt[(36*x^6 + Cosh[6*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^4] - x^2*Sqrt[72*x^2 + (2*Cosh[6*C[1]])/x^4 - (2*Sinh[6*C[1]])/x^4 + (36*2^(2/3)*(-Cosh[3*C[1]] + Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) - (9*2^(1/3)*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^2 + (2*(432*x^6*Cosh[3*C[1]] - Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(x^6*Sqrt[(36*x^6 + Cosh[6*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^4])])/(18*x^2)}, {y[x] -> (-Cosh[3*C[1]] + Sinh[3*C[1]] + x^2*Sqrt[(36*x^6 + Cosh[6*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^4] + x^2*Sqrt[72*x^2 + (2*Cosh[6*C[1]])/x^4 - (2*Sinh[6*C[1]])/x^4 + (36*2^(2/3)*(-Cosh[3*C[1]] + Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) - (9*2^(1/3)*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^2 + (2*(432*x^6*Cosh[3*C[1]] - Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(x^6*Sqrt[(36*x^6 + Cosh[6*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^4])])/(18*x^2)}}
Maple raw input
dsolve(x*diff(y(x),x)^2+2*y(x)*diff(y(x),x)-x = 0, y(x),'implicit')
Maple raw output
[x(_T) = 1/(3*_T^2-1)^(2/3)*_T*_C1, y(_T) = -1/2*(_T^2-1)*_C1/(3*_T^2-1)^(2/3)]