ODE
\[ 2 y'(x)^4-y(x) y'(x)-2=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y\)
Mathematica ✓
cpu = 191.979 (sec), leaf count = 7169
\[\left \{\text {Solve}\left [\frac {x \left (\sqrt {2} 3^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}}+6 \sqrt [12]{3} \sqrt {\frac {-27 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}} y^4-9 \sqrt {2} \sqrt [6]{3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y^3+\left (48 \sqrt {3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}+3\ 3^{2/3} \sqrt {27 y^4+4096} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}-384\ 3^{5/6}\right ) y^2+\sqrt {54 y^4+8192} \left (27 y^2-3 \sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y+16 \left (-\sqrt {27 y^4+4096} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}-128 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}+8 \sqrt [3]{3} \sqrt {27 y^4+4096}\right )}{\left (\sqrt {81 y^4+12288}-9 y^2\right ) \left (-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}\right )}}\right )}{12 \left (\sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}}+\sqrt {2} 3^{5/12} \sqrt {\frac {-27 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}} y^4-9 \sqrt {2} \sqrt [6]{3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y^3+\left (48 \sqrt {3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}+3\ 3^{2/3} \sqrt {27 y^4+4096} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}-384\ 3^{5/6}\right ) y^2+\sqrt {54 y^4+8192} \left (27 y^2-3 \sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y+16 \left (-\sqrt {27 y^4+4096} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}-128 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}+8 \sqrt [3]{3} \sqrt {27 y^4+4096}\right )}{\left (\sqrt {81 y^4+12288}-9 y^2\right ) \left (-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}\right )}}\right )}+\int _1^{y(x)}\frac {1}{\sqrt {-\frac {-9 K[1]^2+\sqrt {81 K[1]^4+12288}+16 \sqrt [3]{27 K[1]^2-3 \sqrt {81 K[1]^4+12288}}}{\left (9 K[1]^2-\sqrt {81 K[1]^4+12288}\right )^{2/3}}}+\sqrt {2} 3^{5/12} \sqrt {\frac {-27 \sqrt [6]{3} \sqrt [3]{9 K[1]^2-\sqrt {81 K[1]^4+12288}} K[1]^4-9 \sqrt {2} \sqrt [6]{3} \left (9 K[1]^2-\sqrt {81 K[1]^4+12288}\right )^{2/3} \sqrt {-\frac {-9 K[1]^2+\sqrt {81 K[1]^4+12288}+16 \sqrt [3]{27 K[1]^2-3 \sqrt {81 K[1]^4+12288}}}{\left (9 K[1]^2-\sqrt {81 K[1]^4+12288}\right )^{2/3}}} K[1]^3+\left (48 \sqrt {3} \left (9 K[1]^2-\sqrt {81 K[1]^4+12288}\right )^{2/3}+3\ 3^{2/3} \sqrt {27 K[1]^4+4096} \sqrt [3]{9 K[1]^2-\sqrt {81 K[1]^4+12288}}-384\ 3^{5/6}\right ) K[1]^2+\sqrt {54 K[1]^4+8192} \left (27 K[1]^2-3 \sqrt {81 K[1]^4+12288}\right )^{2/3} \sqrt {-\frac {-9 K[1]^2+\sqrt {81 K[1]^4+12288}+16 \sqrt [3]{27 K[1]^2-3 \sqrt {81 K[1]^4+12288}}}{\left (9 K[1]^2-\sqrt {81 K[1]^4+12288}\right )^{2/3}}} K[1]+16 \left (-\sqrt {27 K[1]^4+4096} \left (9 K[1]^2-\sqrt {81 K[1]^4+12288}\right )^{2/3}-128 \sqrt [6]{3} \sqrt [3]{9 K[1]^2-\sqrt {81 K[1]^4+12288}}+8 \sqrt [3]{3} \sqrt {27 K[1]^4+4096}\right )}{\left (\sqrt {81 K[1]^4+12288}-9 K[1]^2\right ) \left (-9 K[1]^2+\sqrt {81 K[1]^4+12288}+16 \sqrt [3]{27 K[1]^2-3 \sqrt {81 K[1]^4+12288}}\right )}}}dK[1]=c_1,y(x)\right ],\text {Solve}\left [\int _1^{y(x)}-\frac {1}{\sqrt {-\frac {-9 K[2]^2+\sqrt {81 K[2]^4+12288}+16 \sqrt [3]{27 K[2]^2-3 \sqrt {81 K[2]^4+12288}}}{\left (9 K[2]^2-\sqrt {81 K[2]^4+12288}\right )^{2/3}}}-\sqrt {2} 3^{5/12} \sqrt {\frac {-27 \sqrt [6]{3} \sqrt [3]{9 K[2]^2-\sqrt {81 K[2]^4+12288}} K[2]^4-9 \sqrt {2} \sqrt [6]{3} \left (9 K[2]^2-\sqrt {81 K[2]^4+12288}\right )^{2/3} \sqrt {-\frac {-9 K[2]^2+\sqrt {81 K[2]^4+12288}+16 \sqrt [3]{27 K[2]^2-3 \sqrt {81 K[2]^4+12288}}}{\left (9 K[2]^2-\sqrt {81 K[2]^4+12288}\right )^{2/3}}} K[2]^3+\left (48 \sqrt {3} \left (9 K[2]^2-\sqrt {81 K[2]^4+12288}\right )^{2/3}+3\ 3^{2/3} \sqrt {27 K[2]^4+4096} \sqrt [3]{9 K[2]^2-\sqrt {81 K[2]^4+12288}}-384\ 3^{5/6}\right ) K[2]^2+\sqrt {54 K[2]^4+8192} \left (27 K[2]^2-3 \sqrt {81 K[2]^4+12288}\right )^{2/3} \sqrt {-\frac {-9 K[2]^2+\sqrt {81 K[2]^4+12288}+16 \sqrt [3]{27 K[2]^2-3 \sqrt {81 K[2]^4+12288}}}{\left (9 K[2]^2-\sqrt {81 K[2]^4+12288}\right )^{2/3}}} K[2]+16 \left (-\sqrt {27 K[2]^4+4096} \left (9 K[2]^2-\sqrt {81 K[2]^4+12288}\right )^{2/3}-128 \sqrt [6]{3} \sqrt [3]{9 K[2]^2-\sqrt {81 K[2]^4+12288}}+8 \sqrt [3]{3} \sqrt {27 K[2]^4+4096}\right )}{\left (\sqrt {81 K[2]^4+12288}-9 K[2]^2\right ) \left (-9 K[2]^2+\sqrt {81 K[2]^4+12288}+16 \sqrt [3]{27 K[2]^2-3 \sqrt {81 K[2]^4+12288}}\right )}}}dK[2]=\frac {x \left (\sqrt {2} 3^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}}-6 \sqrt [12]{3} \sqrt {\frac {-27 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}} y^4-9 \sqrt {2} \sqrt [6]{3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y^3+\left (48 \sqrt {3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}+3\ 3^{2/3} \sqrt {27 y^4+4096} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}-384\ 3^{5/6}\right ) y^2+\sqrt {54 y^4+8192} \left (27 y^2-3 \sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y+16 \left (-\sqrt {27 y^4+4096} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}-128 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}+8 \sqrt [3]{3} \sqrt {27 y^4+4096}\right )}{\left (\sqrt {81 y^4+12288}-9 y^2\right ) \left (-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}\right )}}\right )}{12 \left (\sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}}-\sqrt {2} 3^{5/12} \sqrt {\frac {-27 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}} y^4-9 \sqrt {2} \sqrt [6]{3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y^3+\left (48 \sqrt {3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}+3\ 3^{2/3} \sqrt {27 y^4+4096} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}-384\ 3^{5/6}\right ) y^2+\sqrt {54 y^4+8192} \left (27 y^2-3 \sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y+16 \left (-\sqrt {27 y^4+4096} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}-128 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}+8 \sqrt [3]{3} \sqrt {27 y^4+4096}\right )}{\left (\sqrt {81 y^4+12288}-9 y^2\right ) \left (-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}\right )}}\right )}+c_1,y(x)\right ],\text {Solve}\left [\frac {x \left (\sqrt {2} 3^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}}-6 \sqrt [12]{3} \sqrt {-\frac {27 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}} y^4-9 \sqrt {2} \sqrt [6]{3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y^3+\left (-48 \sqrt {3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}-3\ 3^{2/3} \sqrt {27 y^4+4096} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}+384\ 3^{5/6}\right ) y^2+\sqrt {54 y^4+8192} \left (27 y^2-3 \sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y+16 \left (\sqrt {27 y^4+4096} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}+128 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}-8 \sqrt [3]{3} \sqrt {27 y^4+4096}\right )}{\left (\sqrt {81 y^4+12288}-9 y^2\right ) \left (-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}\right )}}\right )}{12 \left (\sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}}-\sqrt {2} 3^{5/12} \sqrt {-\frac {27 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}} y^4-9 \sqrt {2} \sqrt [6]{3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y^3+\left (-48 \sqrt {3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}-3\ 3^{2/3} \sqrt {27 y^4+4096} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}+384\ 3^{5/6}\right ) y^2+\sqrt {54 y^4+8192} \left (27 y^2-3 \sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y+16 \left (\sqrt {27 y^4+4096} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}+128 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}-8 \sqrt [3]{3} \sqrt {27 y^4+4096}\right )}{\left (\sqrt {81 y^4+12288}-9 y^2\right ) \left (-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}\right )}}\right )}+\int _1^{y(x)}-\frac {1}{\sqrt {-\frac {-9 K[3]^2+\sqrt {81 K[3]^4+12288}+16 \sqrt [3]{27 K[3]^2-3 \sqrt {81 K[3]^4+12288}}}{\left (9 K[3]^2-\sqrt {81 K[3]^4+12288}\right )^{2/3}}}-\sqrt {2} 3^{5/12} \sqrt {-\frac {27 \sqrt [6]{3} \sqrt [3]{9 K[3]^2-\sqrt {81 K[3]^4+12288}} K[3]^4-9 \sqrt {2} \sqrt [6]{3} \left (9 K[3]^2-\sqrt {81 K[3]^4+12288}\right )^{2/3} \sqrt {-\frac {-9 K[3]^2+\sqrt {81 K[3]^4+12288}+16 \sqrt [3]{27 K[3]^2-3 \sqrt {81 K[3]^4+12288}}}{\left (9 K[3]^2-\sqrt {81 K[3]^4+12288}\right )^{2/3}}} K[3]^3+\left (-48 \sqrt {3} \left (9 K[3]^2-\sqrt {81 K[3]^4+12288}\right )^{2/3}-3\ 3^{2/3} \sqrt {27 K[3]^4+4096} \sqrt [3]{9 K[3]^2-\sqrt {81 K[3]^4+12288}}+384\ 3^{5/6}\right ) K[3]^2+\sqrt {54 K[3]^4+8192} \left (27 K[3]^2-3 \sqrt {81 K[3]^4+12288}\right )^{2/3} \sqrt {-\frac {-9 K[3]^2+\sqrt {81 K[3]^4+12288}+16 \sqrt [3]{27 K[3]^2-3 \sqrt {81 K[3]^4+12288}}}{\left (9 K[3]^2-\sqrt {81 K[3]^4+12288}\right )^{2/3}}} K[3]+16 \left (\sqrt {27 K[3]^4+4096} \left (9 K[3]^2-\sqrt {81 K[3]^4+12288}\right )^{2/3}+128 \sqrt [6]{3} \sqrt [3]{9 K[3]^2-\sqrt {81 K[3]^4+12288}}-8 \sqrt [3]{3} \sqrt {27 K[3]^4+4096}\right )}{\left (\sqrt {81 K[3]^4+12288}-9 K[3]^2\right ) \left (-9 K[3]^2+\sqrt {81 K[3]^4+12288}+16 \sqrt [3]{27 K[3]^2-3 \sqrt {81 K[3]^4+12288}}\right )}}}dK[3]=c_1,y(x)\right ],\text {Solve}\left [\int _1^{y(x)}\frac {1}{\sqrt {-\frac {-9 K[4]^2+\sqrt {81 K[4]^4+12288}+16 \sqrt [3]{27 K[4]^2-3 \sqrt {81 K[4]^4+12288}}}{\left (9 K[4]^2-\sqrt {81 K[4]^4+12288}\right )^{2/3}}}+\sqrt {2} 3^{5/12} \sqrt {-\frac {27 \sqrt [6]{3} \sqrt [3]{9 K[4]^2-\sqrt {81 K[4]^4+12288}} K[4]^4-9 \sqrt {2} \sqrt [6]{3} \left (9 K[4]^2-\sqrt {81 K[4]^4+12288}\right )^{2/3} \sqrt {-\frac {-9 K[4]^2+\sqrt {81 K[4]^4+12288}+16 \sqrt [3]{27 K[4]^2-3 \sqrt {81 K[4]^4+12288}}}{\left (9 K[4]^2-\sqrt {81 K[4]^4+12288}\right )^{2/3}}} K[4]^3+\left (-48 \sqrt {3} \left (9 K[4]^2-\sqrt {81 K[4]^4+12288}\right )^{2/3}-3\ 3^{2/3} \sqrt {27 K[4]^4+4096} \sqrt [3]{9 K[4]^2-\sqrt {81 K[4]^4+12288}}+384\ 3^{5/6}\right ) K[4]^2+\sqrt {54 K[4]^4+8192} \left (27 K[4]^2-3 \sqrt {81 K[4]^4+12288}\right )^{2/3} \sqrt {-\frac {-9 K[4]^2+\sqrt {81 K[4]^4+12288}+16 \sqrt [3]{27 K[4]^2-3 \sqrt {81 K[4]^4+12288}}}{\left (9 K[4]^2-\sqrt {81 K[4]^4+12288}\right )^{2/3}}} K[4]+16 \left (\sqrt {27 K[4]^4+4096} \left (9 K[4]^2-\sqrt {81 K[4]^4+12288}\right )^{2/3}+128 \sqrt [6]{3} \sqrt [3]{9 K[4]^2-\sqrt {81 K[4]^4+12288}}-8 \sqrt [3]{3} \sqrt {27 K[4]^4+4096}\right )}{\left (\sqrt {81 K[4]^4+12288}-9 K[4]^2\right ) \left (-9 K[4]^2+\sqrt {81 K[4]^4+12288}+16 \sqrt [3]{27 K[4]^2-3 \sqrt {81 K[4]^4+12288}}\right )}}}dK[4]=\frac {x \left (\sqrt {2} 3^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}}+6 \sqrt [12]{3} \sqrt {-\frac {27 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}} y^4-9 \sqrt {2} \sqrt [6]{3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y^3+\left (-48 \sqrt {3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}-3\ 3^{2/3} \sqrt {27 y^4+4096} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}+384\ 3^{5/6}\right ) y^2+\sqrt {54 y^4+8192} \left (27 y^2-3 \sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y+16 \left (\sqrt {27 y^4+4096} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}+128 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}-8 \sqrt [3]{3} \sqrt {27 y^4+4096}\right )}{\left (\sqrt {81 y^4+12288}-9 y^2\right ) \left (-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}\right )}}\right )}{12 \left (\sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}}+\sqrt {2} 3^{5/12} \sqrt {-\frac {27 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}} y^4-9 \sqrt {2} \sqrt [6]{3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y^3+\left (-48 \sqrt {3} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}-3\ 3^{2/3} \sqrt {27 y^4+4096} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}+384\ 3^{5/6}\right ) y^2+\sqrt {54 y^4+8192} \left (27 y^2-3 \sqrt {81 y^4+12288}\right )^{2/3} \sqrt {-\frac {-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}}{\left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}}} y+16 \left (\sqrt {27 y^4+4096} \left (9 y^2-\sqrt {81 y^4+12288}\right )^{2/3}+128 \sqrt [6]{3} \sqrt [3]{9 y^2-\sqrt {81 y^4+12288}}-8 \sqrt [3]{3} \sqrt {27 y^4+4096}\right )}{\left (\sqrt {81 y^4+12288}-9 y^2\right ) \left (-9 y^2+\sqrt {81 y^4+12288}+16 \sqrt [3]{27 y^2-3 \sqrt {81 y^4+12288}}\right )}}\right )}+c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 19.121 (sec), leaf count = 513
\[\left [y \left (x \right ) = -\frac {\sqrt {-6 \textit {\_C1}^{3}+18 \textit {\_C1}^{2} x -18 x^{2} \textit {\_C1} +6 x^{3}+216 \textit {\_C1} -216 x -6 \sqrt {\textit {\_C1}^{6}-6 \textit {\_C1}^{5} x +15 \textit {\_C1}^{4} x^{2}-20 \textit {\_C1}^{3} x^{3}+15 \textit {\_C1}^{2} x^{4}-6 \textit {\_C1} \,x^{5}+x^{6}+36 \textit {\_C1}^{4}-144 \textit {\_C1}^{3} x +216 \textit {\_C1}^{2} x^{2}-144 \textit {\_C1} \,x^{3}+36 x^{4}+432 \textit {\_C1}^{2}-864 \textit {\_C1} x +432 x^{2}+1728}}}{9}, y \left (x \right ) = \frac {\sqrt {-6 \textit {\_C1}^{3}+18 \textit {\_C1}^{2} x -18 x^{2} \textit {\_C1} +6 x^{3}+216 \textit {\_C1} -216 x -6 \sqrt {\textit {\_C1}^{6}-6 \textit {\_C1}^{5} x +15 \textit {\_C1}^{4} x^{2}-20 \textit {\_C1}^{3} x^{3}+15 \textit {\_C1}^{2} x^{4}-6 \textit {\_C1} \,x^{5}+x^{6}+36 \textit {\_C1}^{4}-144 \textit {\_C1}^{3} x +216 \textit {\_C1}^{2} x^{2}-144 \textit {\_C1} \,x^{3}+36 x^{4}+432 \textit {\_C1}^{2}-864 \textit {\_C1} x +432 x^{2}+1728}}}{9}, y \left (x \right ) = -\frac {\sqrt {-6 \textit {\_C1}^{3}+18 \textit {\_C1}^{2} x -18 x^{2} \textit {\_C1} +6 x^{3}+216 \textit {\_C1} -216 x +6 \sqrt {\textit {\_C1}^{6}-6 \textit {\_C1}^{5} x +15 \textit {\_C1}^{4} x^{2}-20 \textit {\_C1}^{3} x^{3}+15 \textit {\_C1}^{2} x^{4}-6 \textit {\_C1} \,x^{5}+x^{6}+36 \textit {\_C1}^{4}-144 \textit {\_C1}^{3} x +216 \textit {\_C1}^{2} x^{2}-144 \textit {\_C1} \,x^{3}+36 x^{4}+432 \textit {\_C1}^{2}-864 \textit {\_C1} x +432 x^{2}+1728}}}{9}, y \left (x \right ) = \frac {\sqrt {-6 \textit {\_C1}^{3}+18 \textit {\_C1}^{2} x -18 x^{2} \textit {\_C1} +6 x^{3}+216 \textit {\_C1} -216 x +6 \sqrt {\textit {\_C1}^{6}-6 \textit {\_C1}^{5} x +15 \textit {\_C1}^{4} x^{2}-20 \textit {\_C1}^{3} x^{3}+15 \textit {\_C1}^{2} x^{4}-6 \textit {\_C1} \,x^{5}+x^{6}+36 \textit {\_C1}^{4}-144 \textit {\_C1}^{3} x +216 \textit {\_C1}^{2} x^{2}-144 \textit {\_C1} \,x^{3}+36 x^{4}+432 \textit {\_C1}^{2}-864 \textit {\_C1} x +432 x^{2}+1728}}}{9}\right ]\] Mathematica raw input
DSolve[-2 - y[x]*y'[x] + 2*y'[x]^4 == 0,y[x],x]
Mathematica raw output
{Solve[(x*(Sqrt[2]*3^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] + 6*3^(1/12)*Sqrt[(-27*3^(1/6)*y^4*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + y*Sqrt[8192 + 54*y^4]*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] - 9*Sqrt[2]*3^(1/6)*y^3*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] + y^2*(-384*3^(5/6) + 3*3^(2/3)*Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + 48*Sqrt[3]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)) + 16*(8*3^(1/3)*Sqrt[4096 + 27*y^4] - 128*3^(1/6)*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) - Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)))/((-9*y^2 + Sqrt[12288 + 81*y^4])*(-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3)))]))/(12*(Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] + Sqrt[2]*3^(5/12)*Sqrt[(-27*3^(1/6)*y^4*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + y*Sqrt[8192 + 54*y^4]*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] - 9*Sqrt[2]*3^(1/6)*y^3*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] + y^2*(-384*3^(5/6) + 3*3^(2/3)*Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + 48*Sqrt[3]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)) + 16*(8*3^(1/3)*Sqrt[4096 + 27*y^4] - 128*3^(1/6)*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) - Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)))/((-9*y^2 + Sqrt[12288 + 81*y^4])*(-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3)))])) + Inactive[Integrate][(Sqrt[-((-9*K[1]^2 + Sqrt[12288 + 81*K[1]^4] + 16*(27*K[1]^2 - 3*Sqrt[12288 + 81*K[1]^4])^(1/3))/(9*K[1]^2 - Sqrt[12288 + 81*K[1]^4])^(2/3))] + Sqrt[2]*3^(5/12)*Sqrt[(-27*3^(1/6)*K[1]^4*(9*K[1]^2 - Sqrt[12288 + 81*K[1]^4])^(1/3) + K[1]*Sqrt[8192 + 54*K[1]^4]*(27*K[1]^2 - 3*Sqrt[12288 + 81*K[1]^4])^(2/3)*Sqrt[-((-9*K[1]^2 + Sqrt[12288 + 81*K[1]^4] + 16*(27*K[1]^2 - 3*Sqrt[12288 + 81*K[1]^4])^(1/3))/(9*K[1]^2 - Sqrt[12288 + 81*K[1]^4])^(2/3))] - 9*Sqrt[2]*3^(1/6)*K[1]^3*(9*K[1]^2 - Sqrt[12288 + 81*K[1]^4])^(2/3)*Sqrt[-((-9*K[1]^2 + Sqrt[12288 + 81*K[1]^4] + 16*(27*K[1]^2 - 3*Sqrt[12288 + 81*K[1]^4])^(1/3))/(9*K[1]^2 - Sqrt[12288 + 81*K[1]^4])^(2/3))] + K[1]^2*(-384*3^(5/6) + 3*3^(2/3)*Sqrt[4096 + 27*K[1]^4]*(9*K[1]^2 - Sqrt[12288 + 81*K[1]^4])^(1/3) + 48*Sqrt[3]*(9*K[1]^2 - Sqrt[12288 + 81*K[1]^4])^(2/3)) + 16*(8*3^(1/3)*Sqrt[4096 + 27*K[1]^4] - 128*3^(1/6)*(9*K[1]^2 - Sqrt[12288 + 81*K[1]^4])^(1/3) - Sqrt[4096 + 27*K[1]^4]*(9*K[1]^2 - Sqrt[12288 + 81*K[1]^4])^(2/3)))/((-9*K[1]^2 + Sqrt[12288 + 81*K[1]^4])*(-9*K[1]^2 + Sqrt[12288 + 81*K[1]^4] + 16*(27*K[1]^2 - 3*Sqrt[12288 + 81*K[1]^4])^(1/3)))])^(-1), {K[1], 1, y[x]}] == C[1], y[x]], Solve[Inactive[Integrate][-(Sqrt[-((-9*K[2]^2 + Sqrt[12288 + 81*K[2]^4] + 16*(27*K[2]^2 - 3*Sqrt[12288 + 81*K[2]^4])^(1/3))/(9*K[2]^2 - Sqrt[12288 + 81*K[2]^4])^(2/3))] - Sqrt[2]*3^(5/12)*Sqrt[(-27*3^(1/6)*K[2]^4*(9*K[2]^2 - Sqrt[12288 + 81*K[2]^4])^(1/3) + K[2]*Sqrt[8192 + 54*K[2]^4]*(27*K[2]^2 - 3*Sqrt[12288 + 81*K[2]^4])^(2/3)*Sqrt[-((-9*K[2]^2 + Sqrt[12288 + 81*K[2]^4] + 16*(27*K[2]^2 - 3*Sqrt[12288 + 81*K[2]^4])^(1/3))/(9*K[2]^2 - Sqrt[12288 + 81*K[2]^4])^(2/3))] - 9*Sqrt[2]*3^(1/6)*K[2]^3*(9*K[2]^2 - Sqrt[12288 + 81*K[2]^4])^(2/3)*Sqrt[-((-9*K[2]^2 + Sqrt[12288 + 81*K[2]^4] + 16*(27*K[2]^2 - 3*Sqrt[12288 + 81*K[2]^4])^(1/3))/(9*K[2]^2 - Sqrt[12288 + 81*K[2]^4])^(2/3))] + K[2]^2*(-384*3^(5/6) + 3*3^(2/3)*Sqrt[4096 + 27*K[2]^4]*(9*K[2]^2 - Sqrt[12288 + 81*K[2]^4])^(1/3) + 48*Sqrt[3]*(9*K[2]^2 - Sqrt[12288 + 81*K[2]^4])^(2/3)) + 16*(8*3^(1/3)*Sqrt[4096 + 27*K[2]^4] - 128*3^(1/6)*(9*K[2]^2 - Sqrt[12288 + 81*K[2]^4])^(1/3) - Sqrt[4096 + 27*K[2]^4]*(9*K[2]^2 - Sqrt[12288 + 81*K[2]^4])^(2/3)))/((-9*K[2]^2 + Sqrt[12288 + 81*K[2]^4])*(-9*K[2]^2 + Sqrt[12288 + 81*K[2]^4] + 16*(27*K[2]^2 - 3*Sqrt[12288 + 81*K[2]^4])^(1/3)))])^(-1), {K[2], 1, y[x]}] == (x*(Sqrt[2]*3^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] - 6*3^(1/12)*Sqrt[(-27*3^(1/6)*y^4*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + y*Sqrt[8192 + 54*y^4]*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] - 9*Sqrt[2]*3^(1/6)*y^3*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] + y^2*(-384*3^(5/6) + 3*3^(2/3)*Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + 48*Sqrt[3]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)) + 16*(8*3^(1/3)*Sqrt[4096 + 27*y^4] - 128*3^(1/6)*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) - Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)))/((-9*y^2 + Sqrt[12288 + 81*y^4])*(-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3)))]))/(12*(Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] - Sqrt[2]*3^(5/12)*Sqrt[(-27*3^(1/6)*y^4*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + y*Sqrt[8192 + 54*y^4]*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] - 9*Sqrt[2]*3^(1/6)*y^3*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] + y^2*(-384*3^(5/6) + 3*3^(2/3)*Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + 48*Sqrt[3]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)) + 16*(8*3^(1/3)*Sqrt[4096 + 27*y^4] - 128*3^(1/6)*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) - Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)))/((-9*y^2 + Sqrt[12288 + 81*y^4])*(-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3)))])) + C[1], y[x]], Solve[(x*(Sqrt[2]*3^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] - 6*3^(1/12)*Sqrt[-((27*3^(1/6)*y^4*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + y*Sqrt[8192 + 54*y^4]*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] - 9*Sqrt[2]*3^(1/6)*y^3*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] + y^2*(384*3^(5/6) - 3*3^(2/3)*Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) - 48*Sqrt[3]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)) + 16*(-8*3^(1/3)*Sqrt[4096 + 27*y^4] + 128*3^(1/6)*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)))/((-9*y^2 + Sqrt[12288 + 81*y^4])*(-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))))]))/(12*(Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] - Sqrt[2]*3^(5/12)*Sqrt[-((27*3^(1/6)*y^4*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + y*Sqrt[8192 + 54*y^4]*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] - 9*Sqrt[2]*3^(1/6)*y^3*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] + y^2*(384*3^(5/6) - 3*3^(2/3)*Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) - 48*Sqrt[3]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)) + 16*(-8*3^(1/3)*Sqrt[4096 + 27*y^4] + 128*3^(1/6)*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)))/((-9*y^2 + Sqrt[12288 + 81*y^4])*(-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))))])) + Inactive[Integrate][-(Sqrt[-((-9*K[3]^2 + Sqrt[12288 + 81*K[3]^4] + 16*(27*K[3]^2 - 3*Sqrt[12288 + 81*K[3]^4])^(1/3))/(9*K[3]^2 - Sqrt[12288 + 81*K[3]^4])^(2/3))] - Sqrt[2]*3^(5/12)*Sqrt[-((27*3^(1/6)*K[3]^4*(9*K[3]^2 - Sqrt[12288 + 81*K[3]^4])^(1/3) + K[3]*Sqrt[8192 + 54*K[3]^4]*(27*K[3]^2 - 3*Sqrt[12288 + 81*K[3]^4])^(2/3)*Sqrt[-((-9*K[3]^2 + Sqrt[12288 + 81*K[3]^4] + 16*(27*K[3]^2 - 3*Sqrt[12288 + 81*K[3]^4])^(1/3))/(9*K[3]^2 - Sqrt[12288 + 81*K[3]^4])^(2/3))] - 9*Sqrt[2]*3^(1/6)*K[3]^3*(9*K[3]^2 - Sqrt[12288 + 81*K[3]^4])^(2/3)*Sqrt[-((-9*K[3]^2 + Sqrt[12288 + 81*K[3]^4] + 16*(27*K[3]^2 - 3*Sqrt[12288 + 81*K[3]^4])^(1/3))/(9*K[3]^2 - Sqrt[12288 + 81*K[3]^4])^(2/3))] + K[3]^2*(384*3^(5/6) - 3*3^(2/3)*Sqrt[4096 + 27*K[3]^4]*(9*K[3]^2 - Sqrt[12288 + 81*K[3]^4])^(1/3) - 48*Sqrt[3]*(9*K[3]^2 - Sqrt[12288 + 81*K[3]^4])^(2/3)) + 16*(-8*3^(1/3)*Sqrt[4096 + 27*K[3]^4] + 128*3^(1/6)*(9*K[3]^2 - Sqrt[12288 + 81*K[3]^4])^(1/3) + Sqrt[4096 + 27*K[3]^4]*(9*K[3]^2 - Sqrt[12288 + 81*K[3]^4])^(2/3)))/((-9*K[3]^2 + Sqrt[12288 + 81*K[3]^4])*(-9*K[3]^2 + Sqrt[12288 + 81*K[3]^4] + 16*(27*K[3]^2 - 3*Sqrt[12288 + 81*K[3]^4])^(1/3))))])^(-1), {K[3], 1, y[x]}] == C[1], y[x]], Solve[Inactive[Integrate][(Sqrt[-((-9*K[4]^2 + Sqrt[12288 + 81*K[4]^4] + 16*(27*K[4]^2 - 3*Sqrt[12288 + 81*K[4]^4])^(1/3))/(9*K[4]^2 - Sqrt[12288 + 81*K[4]^4])^(2/3))] + Sqrt[2]*3^(5/12)*Sqrt[-((27*3^(1/6)*K[4]^4*(9*K[4]^2 - Sqrt[12288 + 81*K[4]^4])^(1/3) + K[4]*Sqrt[8192 + 54*K[4]^4]*(27*K[4]^2 - 3*Sqrt[12288 + 81*K[4]^4])^(2/3)*Sqrt[-((-9*K[4]^2 + Sqrt[12288 + 81*K[4]^4] + 16*(27*K[4]^2 - 3*Sqrt[12288 + 81*K[4]^4])^(1/3))/(9*K[4]^2 - Sqrt[12288 + 81*K[4]^4])^(2/3))] - 9*Sqrt[2]*3^(1/6)*K[4]^3*(9*K[4]^2 - Sqrt[12288 + 81*K[4]^4])^(2/3)*Sqrt[-((-9*K[4]^2 + Sqrt[12288 + 81*K[4]^4] + 16*(27*K[4]^2 - 3*Sqrt[12288 + 81*K[4]^4])^(1/3))/(9*K[4]^2 - Sqrt[12288 + 81*K[4]^4])^(2/3))] + K[4]^2*(384*3^(5/6) - 3*3^(2/3)*Sqrt[4096 + 27*K[4]^4]*(9*K[4]^2 - Sqrt[12288 + 81*K[4]^4])^(1/3) - 48*Sqrt[3]*(9*K[4]^2 - Sqrt[12288 + 81*K[4]^4])^(2/3)) + 16*(-8*3^(1/3)*Sqrt[4096 + 27*K[4]^4] + 128*3^(1/6)*(9*K[4]^2 - Sqrt[12288 + 81*K[4]^4])^(1/3) + Sqrt[4096 + 27*K[4]^4]*(9*K[4]^2 - Sqrt[12288 + 81*K[4]^4])^(2/3)))/((-9*K[4]^2 + Sqrt[12288 + 81*K[4]^4])*(-9*K[4]^2 + Sqrt[12288 + 81*K[4]^4] + 16*(27*K[4]^2 - 3*Sqrt[12288 + 81*K[4]^4])^(1/3))))])^(-1), {K[4], 1, y[x]}] == (x*(Sqrt[2]*3^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] + 6*3^(1/12)*Sqrt[-((27*3^(1/6)*y^4*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + y*Sqrt[8192 + 54*y^4]*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] - 9*Sqrt[2]*3^(1/6)*y^3*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] + y^2*(384*3^(5/6) - 3*3^(2/3)*Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) - 48*Sqrt[3]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)) + 16*(-8*3^(1/3)*Sqrt[4096 + 27*y^4] + 128*3^(1/6)*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)))/((-9*y^2 + Sqrt[12288 + 81*y^4])*(-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))))]))/(12*(Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] + Sqrt[2]*3^(5/12)*Sqrt[-((27*3^(1/6)*y^4*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + y*Sqrt[8192 + 54*y^4]*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] - 9*Sqrt[2]*3^(1/6)*y^3*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)*Sqrt[-((-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))/(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3))] + y^2*(384*3^(5/6) - 3*3^(2/3)*Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) - 48*Sqrt[3]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)) + 16*(-8*3^(1/3)*Sqrt[4096 + 27*y^4] + 128*3^(1/6)*(9*y^2 - Sqrt[12288 + 81*y^4])^(1/3) + Sqrt[4096 + 27*y^4]*(9*y^2 - Sqrt[12288 + 81*y^4])^(2/3)))/((-9*y^2 + Sqrt[12288 + 81*y^4])*(-9*y^2 + Sqrt[12288 + 81*y^4] + 16*(27*y^2 - 3*Sqrt[12288 + 81*y^4])^(1/3))))])) + C[1], y[x]]}
Maple raw input
dsolve(2*diff(y(x),x)^4-y(x)*diff(y(x),x)-2 = 0, y(x))
Maple raw output
[y(x) = -1/9*(-6*_C1^3+18*_C1^2*x-18*x^2*_C1+6*x^3+216*_C1-216*x-6*(_C1^6-6*_C1^5*x+15*_C1^4*x^2-20*_C1^3*x^3+15*_C1^2*x^4-6*_C1*x^5+x^6+36*_C1^4-144*_C1^3*x+216*_C1^2*x^2-144*_C1*x^3+36*x^4+432*_C1^2-864*_C1*x+432*x^2+1728)^(1/2))^(1/2), y(x) = 1/9*(-6*_C1^3+18*_C1^2*x-18*x^2*_C1+6*x^3+216*_C1-216*x-6*(_C1^6-6*_C1^5*x+15*_C1^4*x^2-20*_C1^3*x^3+15*_C1^2*x^4-6*_C1*x^5+x^6+36*_C1^4-144*_C1^3*x+216*_C1^2*x^2-144*_C1*x^3+36*x^4+432*_C1^2-864*_C1*x+432*x^2+1728)^(1/2))^(1/2), y(x) = -1/9*(-6*_C1^3+18*_C1^2*x-18*x^2*_C1+6*x^3+216*_C1-216*x+6*(_C1^6-6*_C1^5*x+15*_C1^4*x^2-20*_C1^3*x^3+15*_C1^2*x^4-6*_C1*x^5+x^6+36*_C1^4-144*_C1^3*x+216*_C1^2*x^2-144*_C1*x^3+36*x^4+432*_C1^2-864*_C1*x+432*x^2+1728)^(1/2))^(1/2), y(x) = 1/9*(-6*_C1^3+18*_C1^2*x-18*x^2*_C1+6*x^3+216*_C1-216*x+6*(_C1^6-6*_C1^5*x+15*_C1^4*x^2-20*_C1^3*x^3+15*_C1^2*x^4-6*_C1*x^5+x^6+36*_C1^4-144*_C1^3*x+216*_C1^2*x^2-144*_C1*x^3+36*x^4+432*_C1^2-864*_C1*x+432*x^2+1728)^(1/2))^(1/2)]