4.18.19 \(a+x y'(x)^2-2 y(x) y'(x)=0\)

ODE
\[ a+x y'(x)^2-2 y(x) y'(x)=0 \] ODE Classification

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

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica
cpu = 1.70487 (sec), leaf count = 9391

\[\left \{\left \{y(x)\to -\frac {1}{4} \sqrt {\frac {9 a^4 \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) x^8}{\sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}}}-\frac {32 a^3 x^5}{\sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}}}+3 a^2 \cosh \left (3 c_1\right ) x^4-3 a^2 \sinh \left (3 c_1\right ) x^4+\frac {32 a^2 \left (\cosh \left (3 c_1\right )+\sinh \left (3 c_1\right )\right ) x^2}{\sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}}}+16 a x+\left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}}}\right \},\left \{y(x)\to \frac {1}{4} \sqrt {\frac {9 a^4 \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) x^8}{\sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}}}-\frac {32 a^3 x^5}{\sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}}}+3 a^2 \cosh \left (3 c_1\right ) x^4-3 a^2 \sinh \left (3 c_1\right ) x^4+\frac {32 a^2 \left (\cosh \left (3 c_1\right )+\sinh \left (3 c_1\right )\right ) x^2}{\sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}}}+16 a x+\left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}}}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {9 i \sqrt {3} a^4 \cosh \left (3 c_1\right ) x^8-9 a^4 \cosh \left (3 c_1\right ) x^8-9 i \sqrt {3} a^4 \sinh \left (3 c_1\right ) x^8+9 a^4 \sinh \left (3 c_1\right ) x^8-32 i \sqrt {3} a^3 x^5+32 a^3 x^5+6 a^2 \cosh \left (3 c_1\right ) \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}} x^4-6 a^2 \sinh \left (3 c_1\right ) \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}} x^4+32 i \sqrt {3} a^2 \cosh \left (3 c_1\right ) x^2-32 a^2 \cosh \left (3 c_1\right ) x^2+32 i \sqrt {3} a^2 \sinh \left (3 c_1\right ) x^2-32 a^2 \sinh \left (3 c_1\right ) x^2+32 a \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}} x-i \sqrt {3} \cosh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}-\cosh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}+i \sqrt {3} \sinh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}+\sinh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}}{\sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}}}}}{4 \sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {9 i \sqrt {3} a^4 \cosh \left (3 c_1\right ) x^8-9 a^4 \cosh \left (3 c_1\right ) x^8-9 i \sqrt {3} a^4 \sinh \left (3 c_1\right ) x^8+9 a^4 \sinh \left (3 c_1\right ) x^8-32 i \sqrt {3} a^3 x^5+32 a^3 x^5+6 a^2 \cosh \left (3 c_1\right ) \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}} x^4-6 a^2 \sinh \left (3 c_1\right ) \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}} x^4+32 i \sqrt {3} a^2 \cosh \left (3 c_1\right ) x^2-32 a^2 \cosh \left (3 c_1\right ) x^2+32 i \sqrt {3} a^2 \sinh \left (3 c_1\right ) x^2-32 a^2 \sinh \left (3 c_1\right ) x^2+32 a \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}} x-i \sqrt {3} \cosh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}-\cosh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}+i \sqrt {3} \sinh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}+\sinh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}}{\sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}}}}}{4 \sqrt {2}}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {-9 i \sqrt {3} a^4 \cosh \left (3 c_1\right ) x^8-9 a^4 \cosh \left (3 c_1\right ) x^8+9 i \sqrt {3} a^4 \sinh \left (3 c_1\right ) x^8+9 a^4 \sinh \left (3 c_1\right ) x^8+32 i \sqrt {3} a^3 x^5+32 a^3 x^5+6 a^2 \cosh \left (3 c_1\right ) \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}} x^4-6 a^2 \sinh \left (3 c_1\right ) \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}} x^4-32 i \sqrt {3} a^2 \cosh \left (3 c_1\right ) x^2-32 a^2 \cosh \left (3 c_1\right ) x^2-32 i \sqrt {3} a^2 \sinh \left (3 c_1\right ) x^2-32 a^2 \sinh \left (3 c_1\right ) x^2+32 a \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}} x+i \sqrt {3} \cosh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}-\cosh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}-i \sqrt {3} \sinh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}+\sinh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}}{\sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}}}}}{4 \sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {-9 i \sqrt {3} a^4 \cosh \left (3 c_1\right ) x^8-9 a^4 \cosh \left (3 c_1\right ) x^8+9 i \sqrt {3} a^4 \sinh \left (3 c_1\right ) x^8+9 a^4 \sinh \left (3 c_1\right ) x^8+32 i \sqrt {3} a^3 x^5+32 a^3 x^5+6 a^2 \cosh \left (3 c_1\right ) \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}} x^4-6 a^2 \sinh \left (3 c_1\right ) \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}} x^4-32 i \sqrt {3} a^2 \cosh \left (3 c_1\right ) x^2-32 a^2 \cosh \left (3 c_1\right ) x^2-32 i \sqrt {3} a^2 \sinh \left (3 c_1\right ) x^2-32 a^2 \sinh \left (3 c_1\right ) x^2+32 a \sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}} x+i \sqrt {3} \cosh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}-\cosh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}-i \sqrt {3} \sinh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}+\sinh \left (3 c_1\right ) \left (27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}\right ){}^{2/3}}{\sqrt [3]{27 a^6 x^{12}-144 a^5 \cosh \left (3 c_1\right ) x^9-144 a^5 \sinh \left (3 c_1\right ) x^9+272 a^4 \cosh \left (6 c_1\right ) x^6+272 a^4 \sinh \left (6 c_1\right ) x^6-256 a^3 \cosh \left (9 c_1\right ) x^3-256 a^3 \sinh \left (9 c_1\right ) x^3+128 a^2 \cosh \left (12 c_1\right )+128 a^2 \sinh \left (12 c_1\right )+64 \sqrt {-a^4 \left (\left (a x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\left (a x^3-2\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (a x^3+2\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^2 \left (\cosh \left (\frac {33 c_1}{2}\right )+\sinh \left (\frac {33 c_1}{2}\right )\right )}}}}}{4 \sqrt {2}}\right \}\right \}\]

Maple
cpu = 0.017 (sec), leaf count = 33

\[ \left \{ [x \left ({\it \_T} \right ) ={\it \_T}\, \left ({\frac {a}{3\,{{\it \_T}}^{3}}}+{\it \_C1} \right ) ,y \left ({\it \_T} \right ) ={\frac {3\,{\it \_C1}\,{{\it \_T}}^{3}+4\,a}{6\,{\it \_T}}}] \right \} \] Mathematica raw input

DSolve[a - 2*y[x]*y'[x] + x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -Sqrt[16*a*x + 3*a^2*x^4*Cosh[3*C[1]] - 3*a^2*x^4*Sinh[3*C[1]] - (32*a
^3*x^5)/(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256
*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a
^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt
[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a
*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Si
nh[(33*C[1])/2]))])^(1/3) + (9*a^4*x^8*(Cosh[3*C[1]] - Sinh[3*C[1]]))/(27*a^6*x^
12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[
1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]
] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^
3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1]
)/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))]
)^(1/3) + (32*a^2*x^2*(Cosh[3*C[1]] + Sinh[3*C[1]]))/(27*a^6*x^12 - 144*a^5*x^9*
Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cos
h[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*S
inh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/
2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3
)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(1/3) + (Cosh[3
*C[1]] - Sinh[3*C[1]])*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cos
h[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[
3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*
C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])
/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(3
3*C[1])/2] + Sinh[(33*C[1])/2]))])^(1/3)]/4}, {y[x] -> Sqrt[16*a*x + 3*a^2*x^4*C
osh[3*C[1]] - 3*a^2*x^4*Sinh[3*C[1]] - (32*a^3*x^5)/(27*a^6*x^12 - 144*a^5*x^9*C
osh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh
[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Si
nh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2
] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)
*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(1/3) + (9*a^4*x
^8*(Cosh[3*C[1]] - Sinh[3*C[1]]))/(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*
a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^
5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a
^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Si
nh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])
^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(1/3) + (32*a^2*x^2*(Cosh[3*C[1]] 
+ Sinh[3*C[1]]))/(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[
1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]
] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] 
+ 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3
*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1]
)/2] + Sinh[(33*C[1])/2]))])^(1/3) + (Cosh[3*C[1]] - Sinh[3*C[1]])*(27*a^6*x^12 
- 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]]
 + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] -
 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*
Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2
] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(
1/3)]/4}, {y[x] -> -Sqrt[(32*a^3*x^5 - (32*I)*Sqrt[3]*a^3*x^5 - 32*a^2*x^2*Cosh[
3*C[1]] + (32*I)*Sqrt[3]*a^2*x^2*Cosh[3*C[1]] - 9*a^4*x^8*Cosh[3*C[1]] + (9*I)*S
qrt[3]*a^4*x^8*Cosh[3*C[1]] - 32*a^2*x^2*Sinh[3*C[1]] + (32*I)*Sqrt[3]*a^2*x^2*S
inh[3*C[1]] + 9*a^4*x^8*Sinh[3*C[1]] - (9*I)*Sqrt[3]*a^4*x^8*Sinh[3*C[1]] + 32*a
*x*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*
x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^
6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^
4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)
*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(3
3*C[1])/2]))])^(1/3) + 6*a^2*x^4*Cosh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*
C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[
1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C
[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1
 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[
(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(1/3) - 6*a^2*x^4*Sinh
[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 25
6*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*
a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqr
t[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + 
a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + S
inh[(33*C[1])/2]))])^(1/3) - Cosh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]
] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] 
- 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]]
 + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a
*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C
[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(2/3) - I*Sqrt[3]*Cosh[3*C
[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^
3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*
x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(
a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^
3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[
(33*C[1])/2]))])^(2/3) + Sinh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 
272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 14
4*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 1
28*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3
)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])
/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(2/3) + I*Sqrt[3]*Sinh[3*C[1]]
*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^
3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*
Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*
((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*C
osh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*
C[1])/2]))])^(2/3))/(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6
*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C
[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1
]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2]
)^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C
[1])/2] + Sinh[(33*C[1])/2]))])^(1/3)]/(4*Sqrt[2])}, {y[x] -> Sqrt[(32*a^3*x^5 -
 (32*I)*Sqrt[3]*a^3*x^5 - 32*a^2*x^2*Cosh[3*C[1]] + (32*I)*Sqrt[3]*a^2*x^2*Cosh[
3*C[1]] - 9*a^4*x^8*Cosh[3*C[1]] + (9*I)*Sqrt[3]*a^4*x^8*Cosh[3*C[1]] - 32*a^2*x
^2*Sinh[3*C[1]] + (32*I)*Sqrt[3]*a^2*x^2*Sinh[3*C[1]] + 9*a^4*x^8*Sinh[3*C[1]] -
 (9*I)*Sqrt[3]*a^4*x^8*Sinh[3*C[1]] + 32*a*x*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C
[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1
]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[
1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 
+ a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(
3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(1/3) + 6*a^2*x^4*Cosh[
3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256
*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a
^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt
[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a
*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Si
nh[(33*C[1])/2]))])^(1/3) - 6*a^2*x^4*Sinh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Co
sh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[
12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sin
h[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2]
 - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*
Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(1/3) - Cosh[3*C[
1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3
*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x
^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a
^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3
)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(
33*C[1])/2]))])^(2/3) - I*Sqrt[3]*Cosh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3
*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C
[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*
C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (
1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh
[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(2/3) + Sinh[3*C[1]]*
(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3
*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*S
inh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*(
(-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Co
sh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C
[1])/2]))])^(2/3) + I*Sqrt[3]*Sinh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1
]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]]
 - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]
] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + 
a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*
C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(2/3))/(27*a^6*x^12 - 144
*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 12
8*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*
a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[
(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (
2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(1/3)]
/(4*Sqrt[2])}, {y[x] -> -Sqrt[(32*a^3*x^5 + (32*I)*Sqrt[3]*a^3*x^5 - 32*a^2*x^2*
Cosh[3*C[1]] - (32*I)*Sqrt[3]*a^2*x^2*Cosh[3*C[1]] - 9*a^4*x^8*Cosh[3*C[1]] - (9
*I)*Sqrt[3]*a^4*x^8*Cosh[3*C[1]] - 32*a^2*x^2*Sinh[3*C[1]] - (32*I)*Sqrt[3]*a^2*
x^2*Sinh[3*C[1]] + 9*a^4*x^8*Sinh[3*C[1]] + (9*I)*Sqrt[3]*a^4*x^8*Sinh[3*C[1]] +
 32*a*x*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256
*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a
^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt
[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a
*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Si
nh[(33*C[1])/2]))])^(1/3) + 6*a^2*x^4*Cosh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Co
sh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[
12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sin
h[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2]
 - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*
Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(1/3) - 6*a^2*x^4
*Sinh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]]
 - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] +
 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 6
4*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((
-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2
] + Sinh[(33*C[1])/2]))])^(1/3) - Cosh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3
*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C
[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*
C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (
1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh
[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(2/3) + I*Sqrt[3]*Cos
h[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 2
56*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272
*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sq
rt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 +
 a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + 
Sinh[(33*C[1])/2]))])^(2/3) + Sinh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1
]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]]
 - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]
] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + 
a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*
C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(2/3) - I*Sqrt[3]*Sinh[3*
C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a
^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4
*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-
(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x
^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh
[(33*C[1])/2]))])^(2/3))/(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*C
osh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sin
h[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[1
2*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1
])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[
(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(1/3)]/(4*Sqrt[2])}, {y[x] -> Sqrt[(32*a^3*
x^5 + (32*I)*Sqrt[3]*a^3*x^5 - 32*a^2*x^2*Cosh[3*C[1]] - (32*I)*Sqrt[3]*a^2*x^2*
Cosh[3*C[1]] - 9*a^4*x^8*Cosh[3*C[1]] - (9*I)*Sqrt[3]*a^4*x^8*Cosh[3*C[1]] - 32*
a^2*x^2*Sinh[3*C[1]] - (32*I)*Sqrt[3]*a^2*x^2*Sinh[3*C[1]] + 9*a^4*x^8*Sinh[3*C[
1]] + (9*I)*Sqrt[3]*a^4*x^8*Sinh[3*C[1]] + 32*a*x*(27*a^6*x^12 - 144*a^5*x^9*Cos
h[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[1
2*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh
[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] 
- (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*S
inh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(1/3) + 6*a^2*x^4*
Cosh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] 
- 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 
272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64
*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-
2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2]
 + Sinh[(33*C[1])/2]))])^(1/3) - 6*a^2*x^4*Sinh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x
^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*
Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^
3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1
])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*
x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(1/3) - Cosh
[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 25
6*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*
a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqr
t[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + 
a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + S
inh[(33*C[1])/2]))])^(2/3) + I*Sqrt[3]*Cosh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*C
osh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh
[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Si
nh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2
] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)
*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(2/3) + Sinh[3*C
[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^
3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*
x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(
a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^
3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[
(33*C[1])/2]))])^(2/3) - I*Sqrt[3]*Sinh[3*C[1]]*(27*a^6*x^12 - 144*a^5*x^9*Cosh[
3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]] + 128*a^2*Cosh[12*
C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] - 256*a^3*x^3*Sinh[9
*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*Cosh[(3*C[1])/2] - 
(1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2] - (2 + a*x^3)*Sin
h[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(2/3))/(27*a^6*x^12 
- 144*a^5*x^9*Cosh[3*C[1]] + 272*a^4*x^6*Cosh[6*C[1]] - 256*a^3*x^3*Cosh[9*C[1]]
 + 128*a^2*Cosh[12*C[1]] - 144*a^5*x^9*Sinh[3*C[1]] + 272*a^4*x^6*Sinh[6*C[1]] -
 256*a^3*x^3*Sinh[9*C[1]] + 128*a^2*Sinh[12*C[1]] + 64*Sqrt[-(a^4*((-1 + a*x^3)*
Cosh[(3*C[1])/2] - (1 + a*x^3)*Sinh[(3*C[1])/2])^3*((-2 + a*x^3)*Cosh[(3*C[1])/2
] - (2 + a*x^3)*Sinh[(3*C[1])/2])^2*(Cosh[(33*C[1])/2] + Sinh[(33*C[1])/2]))])^(
1/3)]/(4*Sqrt[2])}}

Maple raw input

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

Maple raw output

[x(_T) = _T*(1/3*a/_T^3+_C1), y(_T) = 1/6/_T*(3*_C1*_T^3+4*a)]