\[ {}{\mathrm e}^{x}+\left ({\mathrm e}^{x} \cot \left (y\right )+2 y \csc \left (y\right )\right ) y^{\prime } = 0 \]

\[ {\mathrm e}^{x} \sin \left (y\right )+y^{2} = c_{1} \] Verified OK.

\[ {}\left (2+x \right ) \sin \left (y\right )+x \cos \left (y\right ) y^{\prime } = 0 \]

\[ -x -2 \ln \left (x \right )-\ln \left (\sin \left (y\right )\right ) = c_{1} \] Verified OK.

\[ {}y+\left (x -2 x^{2} y^{3}\right ) y^{\prime } = 0 \]

\[ -\frac {1}{x y}-y^{2} = c_{1} \] Verified OK.

\[ {}x +3 y^{2}+2 x y y^{\prime } = 0 \]

\[ y^{2} x^{3}+\frac {x^{4}}{4} = c_{1} \] Verified OK.

\[ {}y+\left (2 x -{\mathrm e}^{y} y\right ) y^{\prime } = 0 \]

\[ y^{2} x -\left (y^{2}-2 y+2\right ) {\mathrm e}^{y} = c_{1} \] Verified OK.

\[ {}x^{3}+x y^{3}+3 y^{2} y^{\prime } = 0 \]

\[ \left (y^{3}+x^{2}-2\right ) {\mathrm e}^{\frac {x^{2}}{2}} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {2 y}{x}+\frac {x^{3}}{y}+x \tan \left (\frac {y}{x^{2}}\right ) \]

\[ -\ln \left (\cos \left (\frac {y}{x^{2}}\right ) x^{2}+\sin \left (\frac {y}{x^{2}}\right ) y\right )+2 \ln \left (x \right ) = -\ln \left (x \right )+c_{1} \] Verified OK.

\[ {}y^{\prime \prime } = 1+{y^{\prime }}^{2} \]

\[ -\arctan \left (\frac {1}{\sqrt {-1+c_{1}^{2} {\mathrm e}^{2 y}}}\right ) = x +c_{2} \] Verified OK.

\[ \arctan \left (\frac {1}{\sqrt {-1+c_{1}^{2} {\mathrm e}^{2 y}}}\right ) = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime }+{y^{\prime }}^{2} = 1 \]

\[ \int _{}^{y}\frac {1}{\sqrt {1+\frac {{\mathrm e}^{-2 \textit {\_a}}}{c_{1}^{2}}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ -\frac {{\mathrm e}^{-y} \sqrt {\frac {c_{1}^{2} {\mathrm e}^{2 y}+1}{c_{1}^{2}}}\, \ln \left ({\mathrm e}^{y}+\sqrt {\frac {c_{1}^{2} {\mathrm e}^{2 y}+1}{c_{1}^{2}}}\right )}{\sqrt {\frac {\left (c_{1}^{2} {\mathrm e}^{2 y}+1\right ) {\mathrm e}^{-2 y}}{c_{1}^{2}}}} = x +c_{3} \] Verified OK.

\[ {}\sec \left (x \right ) y^{\prime } = \sec \left (y\right ) \]

\[ -\sin \left (x \right )+\sin \left (y\right ) = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {2 y+x}{2 x -y} \]

\[ \frac {\ln \left (x^{2}+y^{2}\right )}{2}+2 \arctan \left (\frac {x}{y}\right ) = c_{1} \] Verified OK.

\[ {}-\sin \left (x \right ) \sin \left (y\right )+\cos \left (x \right ) \cos \left (y\right ) y^{\prime } = 0 \]

\[ \ln \left (\cos \left (x \right )\right )+\ln \left (\sin \left (y\right )\right ) = c_{1} \] Verified OK.

\[ {}y^{2} y^{\prime } = x \]

\[ \frac {y^{3}}{3}-\frac {x^{2}}{2} = {\frac {17}{2}} \] Verified OK.

\[ {}\csc \left (x \right ) y^{\prime } = \csc \left (y\right ) \]

\[ \cos \left (x \right )-\cos \left (y\right ) = -\cos \left (1\right ) \] Verified OK.

\[ {}y^{\prime } = \frac {x +y}{x -y} \]

\[ \frac {\ln \left (x^{2}+y^{2}\right )}{2}+\arctan \left (\frac {x}{y}\right ) = \frac {\ln \left (2\right )}{2}+\frac {\pi }{4} \] Verified OK.

\[ {}2 x \cos \left (y\right )-x^{2} \sin \left (y\right ) y^{\prime } = 0 \]

\[ -\ln \left (x \right )-\frac {\ln \left (\cos \left (y\right )\right )}{2} = -\frac {\ln \left (\cos \left (1\right )\right )}{2} \] Verified OK.

\[ {}y y^{\prime \prime }+y^{\prime } = 0 \]

\[ \int _{}^{y}\frac {1}{-\ln \left (\textit {\_a} \right )+c_{1}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime \prime }+\sin \left (y\right ) = 0 \]

\[ \frac {2 \sqrt {\frac {\cos \left (y\right )+c_{1}}{c_{1} +1}}\, \operatorname {InverseJacobiAM}\left (\frac {y}{2}, \frac {2}{\sqrt {2 c_{1} +2}}\right )}{\sqrt {2 \cos \left (y\right )+2 c_{1}}} = x +c_{2} \] Verified OK.

\[ -\frac {2 \sqrt {\frac {\cos \left (y\right )+c_{1}}{c_{1} +1}}\, \operatorname {InverseJacobiAM}\left (\frac {y}{2}, \frac {2}{\sqrt {2 c_{1} +2}}\right )}{\sqrt {2 \cos \left (y\right )+2 c_{1}}} = x +c_{3} \] Verified OK.

\[ {}\left (x +y\right )^{2} {y^{\prime }}^{2} = y^{2} \]

\[ \ln \left (\frac {y}{x}\right )-\frac {x}{y}+\ln \left (x \right )-c_{5} = 0 \] Verified OK.

\[ {}\left (4 x -y\right ) {y^{\prime }}^{2}+6 \left (x -y\right ) y^{\prime }+2 x -5 y = 0 \]

\[ -\frac {\left (y+2 x \right )^{2}}{x -y} = c_{4} \] Verified OK.

\[ {}\left (x -y\right )^{2} {y^{\prime }}^{2} = y^{2} \]

\[ \ln \left (\frac {y}{x}\right )+\frac {x}{y}+\ln \left (x \right )-c_{5} = 0 \] Verified OK.

\[ {}\left (x^{2}+y^{2}\right )^{2} {y^{\prime }}^{2} = 4 y^{2} x^{2} \]

\[ {\left (\frac {y \left (y^{2}+3 x^{2}\right )}{x^{3}}\right )}^{\frac {1}{3}} = \frac {c_{6} {\mathrm e}^{c_{5}}}{x} \] Verified OK.

\[ {}\left (x +y\right )^{2} {y^{\prime }}^{2}+\left (2 y^{2}+y x -x^{2}\right ) y^{\prime }+y \left (y-x \right ) = 0 \]

\[ \sqrt {\frac {y \left (y+2 x \right )}{x^{2}}} = \frac {c_{6} {\mathrm e}^{c_{5}}}{x} \] Verified OK.

\[ {}x y \left (x^{2}+y^{2}\right ) \left ({y^{\prime }}^{2}-1\right ) = y^{\prime } \left (x^{4}+y^{2} x^{2}+y^{4}\right ) \]

\[ \left (\frac {y^{2}+2 x^{2}}{x^{2}}\right )^{\frac {1}{4}} \sqrt {\frac {y}{x}} = \frac {c_{5}}{x} \] Verified OK.

\[ {}{y^{\prime }}^{2}-x y^{\prime }-y = 0 \]

\[ y = 0 \] Verified OK.

\[ x = \frac {x}{3}+\frac {\sqrt {x^{2}+4 y}}{3}+\frac {2 c_{1}}{\sqrt {2 x +2 \sqrt {x^{2}+4 y}}} \] Verified OK.

\[ x = \frac {x}{3}-\frac {\sqrt {x^{2}+4 y}}{3}+\frac {2 c_{1}}{\sqrt {2 x -2 \sqrt {x^{2}+4 y}}} \] Verified OK.

\[ {}{y^{\prime }}^{3}+x {y^{\prime }}^{2}-y = 0 \]

\[ y = 0 \] Verified OK.

\[ y = 1+x \] Verified OK.

\[ x = \frac {24 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}}{2}-\frac {27 y}{2}\right ) \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {2}{3}}+96 \left (x +\frac {3}{2}\right ) \left (\left (\sqrt {3}\, \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 y x^{3}}-\frac {x^{6}}{2}+\frac {27 y x^{3}}{2}-\frac {243 y^{2}}{4}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {1}{3}}+\left (-\frac {5 \sqrt {3}\, \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 y x^{3}}}{2}+x^{6}-\frac {81 y x^{3}}{2}+243 y^{2}\right ) x \right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}-27 y\right ) {\left (\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {2}{3}}-2 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {1}{3}}+4 x^{2}-6 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {1}{3}}\right )}^{2}} \] Warning, solution could not be verified

\[ x = \frac {96 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}}{2}-\frac {27 y}{2}\right ) \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {2}{3}}+192 \left (x +\frac {3}{2}\right ) \left (\left (-3 \left (i+\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 y x^{3}}+\frac {\left (1+i \sqrt {3}\right ) \left (x^{6}-27 y x^{3}+\frac {243 y^{2}}{2}\right )}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {1}{3}}+x \left (-\frac {15 \left (i-\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 y x^{3}}}{2}+\left (x^{6}-\frac {81 y x^{3}}{2}+243 y^{2}\right ) \left (i \sqrt {3}-1\right )\right )\right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}-27 y\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {2}{3}}+4 x^{2}+4 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {1}{3}}+\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {2}{3}}+12 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {1}{3}}\right )^{2}} \] Warning, solution could not be verified

\[ x = \frac {96 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}}{2}-\frac {27 y}{2}\right ) \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {2}{3}}-192 \left (x +\frac {3}{2}\right ) \left (\left (-3 \left (i-\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 y x^{3}}+\frac {\left (i \sqrt {3}-1\right ) \left (x^{6}-27 y x^{3}+\frac {243 y^{2}}{2}\right )}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {1}{3}}+\left (-\frac {15 \left (x^{3}-\frac {54 y}{5}\right ) \left (i+\frac {\sqrt {3}}{3}\right ) \sqrt {27 y^{2}-4 y x^{3}}}{2}+\left (x^{6}-\frac {81 y x^{3}}{2}+243 y^{2}\right ) \left (1+i \sqrt {3}\right )\right ) x \right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}-27 y\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {2}{3}}-4 x^{2}-4 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {1}{3}}-\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {2}{3}}-12 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 y x^{3}}\right )^{\frac {1}{3}}\right )^{2}} \] Warning, solution could not be verified

\[ {}{y^{\prime }}^{2}-x y^{\prime }-y = 0 \]

\[ y = 0 \] Verified OK.

\[ x = \frac {x}{3}+\frac {\sqrt {x^{2}+4 y}}{3}+\frac {2 c_{1}}{\sqrt {2 x +2 \sqrt {x^{2}+4 y}}} \] Verified OK.

\[ x = \frac {x}{3}-\frac {\sqrt {x^{2}+4 y}}{3}+\frac {2 c_{1}}{\sqrt {2 x -2 \sqrt {x^{2}+4 y}}} \] Verified OK.

\[ {}2 {y^{\prime }}^{3}+x y^{\prime }-2 y = 0 \]

\[ y = 0 \] Verified OK.

\[ x = \frac {\left (\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}-6 x \right ) \left (\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+c_{1} \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {1}{3}}-6 x \right )}{6 \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}} \] Verified OK.

\[ x = \frac {\left (i \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+6 i \sqrt {3}\, x -\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+6 x \right ) \left (i \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+6 i \sqrt {3}\, x -\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+2 c_{1} \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {1}{3}}+6 x \right )}{24 \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}} \] Warning, solution could not be verified

\[ x = \frac {\left (i \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+6 i \sqrt {3}\, x +\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}-6 x \right ) \left (i \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+6 i \sqrt {3}\, x +\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}-2 c_{1} \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {1}{3}}-6 x \right )}{24 \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}} \] Warning, solution could not be verified

\[ {}2 {y^{\prime }}^{2}+x y^{\prime }-2 y = 0 \]

\[ y = 0 \] Verified OK.

\[ x = -\frac {\left (x -\sqrt {x^{2}+16 y}\right ) \left (-8 \ln \left (2\right )+4 \ln \left (-x +\sqrt {x^{2}+16 y}\right )+c_{1} \right )}{4} \] Verified OK.

\[ x = -\frac {\left (x +\sqrt {x^{2}+16 y}\right ) \left (-8 \ln \left (2\right )+4 \ln \left (-x -\sqrt {x^{2}+16 y}\right )+c_{1} \right )}{4} \] Verified OK.

\[ {}{y^{\prime }}^{3}+2 x y^{\prime }-y = 0 \]

\[ y = 0 \] Verified OK.

\[ x = -\frac {{\left (\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x \right )}^{2}}{48 \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {36 c_{1} \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{{\left (\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x \right )}^{2}} \] Verified OK.

\[ x = \frac {3 {\left (\frac {\left (\sqrt {3}+i\right ) \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{24}+x \left (-i+\sqrt {3}\right )\right )}^{2}}{\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {144 c_{1} \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{{\left (i \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+24 i \sqrt {3}\, x -\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}} \] Verified OK.

\[ x = \frac {3 {\left (\frac {\left (-i+\sqrt {3}\right ) \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{24}+\left (\sqrt {3}+i\right ) x \right )}^{2}}{\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {144 c_{1} \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{{\left (i \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+24 i \sqrt {3}\, x +\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x \right )}^{2}} \] Verified OK.

\[ {}4 x {y^{\prime }}^{2}-3 y y^{\prime }+3 = 0 \]

\[ x = \frac {64 x^{2} \left (64 c_{1} x^{2}+9 y \sqrt {9 y^{2}-48 x}+27 y^{2}-72 x \right )}{\left (3 y+\sqrt {9 y^{2}-48 x}\right )^{4}} \] Verified OK.

\[ x = -\frac {64 x^{2} \left (-64 c_{1} x^{2}+9 y \sqrt {9 y^{2}-48 x}-27 y^{2}+72 x \right )}{\left (-3 y+\sqrt {9 y^{2}-48 x}\right )^{4}} \] Verified OK.

\[ {}{y^{\prime }}^{3}-x y^{\prime }+2 y = 0 \]

\[ y = 0 \] Verified OK.

\[ x = -\frac {\left (\left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+3 x \right ) \left (\left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-c_{1} \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}+3 x \right )}{3 \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}} \] Verified OK.

\[ x = -\frac {\left (-i \sqrt {3}\, \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+3 i \sqrt {3}\, x +\left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+3 x \right ) \left (-i \sqrt {3}\, \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+3 i \sqrt {3}\, x +\left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+2 c_{1} \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}+3 x \right )}{12 \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}} \] Warning, solution could not be verified

\[ x = -\frac {\left (i \sqrt {3}\, \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-3 i \sqrt {3}\, x +\left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+3 x \right ) \left (i \sqrt {3}\, \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-3 i \sqrt {3}\, x +\left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+2 c_{1} \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}+3 x \right )}{12 \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}} \] Warning, solution could not be verified

\[ {}5 {y^{\prime }}^{2}+6 x y^{\prime }-2 y = 0 \]

\[ y = 0 \] Verified OK.

\[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+10 y}}{5}+\frac {125 c_{1}}{\left (-15 x +5 \sqrt {9 x^{2}+10 y}\right )^{\frac {3}{2}}} \] Verified OK.

\[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+10 y}}{5}+\frac {125 c_{1}}{\left (-15 x -5 \sqrt {9 x^{2}+10 y}\right )^{\frac {3}{2}}} \] Verified OK.

\[ {}2 x {y^{\prime }}^{2}+\left (2 x -y\right ) y^{\prime }+1-y = 0 \]

\[ y = 1 \] Verified OK.

\[ x = \frac {32 x^{2} \left (\left (x +\frac {y}{2}+\frac {\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}}{2}\right ) \ln \left (\frac {2 x +y+\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}}{x}\right )+\left (\frac {c_{1}}{2}-\ln \left (2\right )\right ) \sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}+\left (-2 x -y\right ) \ln \left (2\right )+\left (c_{1} +2\right ) x +\frac {c_{1} y}{2}\right )}{\left (2 x +y+\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}\right ) \left (2 x -y-\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}\right )^{2}} \] Verified OK.

\[ x = \frac {32 x^{2} \left (\left (x +\frac {y}{2}-\frac {\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}}{2}\right ) \ln \left (\frac {2 x +y-\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}}{x}\right )+\left (-\frac {c_{1}}{2}+\ln \left (2\right )\right ) \sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}+\left (-2 x -y\right ) \ln \left (2\right )+\left (c_{1} +2\right ) x +\frac {c_{1} y}{2}\right )}{\left (2 x +y-\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}\right ) \left (2 x -y+\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}\right )^{2}} \] Verified OK.

\[ {}5 {y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \]

\[ y = 0 \] Verified OK.

\[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+20 y}}{5}+\frac {1000 c_{1}}{\left (-30 x +10 \sqrt {9 x^{2}+20 y}\right )^{\frac {3}{2}}} \] Verified OK.

\[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+20 y}}{5}+\frac {1000 c_{1}}{\left (-30 x -10 \sqrt {9 x^{2}+20 y}\right )^{\frac {3}{2}}} \] Verified OK.

\[ {}{y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \]

\[ y = 0 \] Verified OK.

\[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+4 y}}{5}+\frac {8 c_{1}}{\left (-6 x +2 \sqrt {9 x^{2}+4 y}\right )^{\frac {3}{2}}} \] Verified OK.

\[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+4 y}}{5}+\frac {8 c_{1}}{\left (-6 x -2 \sqrt {9 x^{2}+4 y}\right )^{\frac {3}{2}}} \] Verified OK.

\[ {}2 y^{\prime \prime } = \sin \left (2 y\right ) \]

\[ -\operatorname {arctanh}\left (\cos \left (y\right )\right ) = x \] Verified OK.

\[ \operatorname {arctanh}\left (\cos \left (y\right )\right ) = x \] Verified OK.

\[ {}2 y^{\prime \prime } = \sin \left (2 y\right ) \]

\[ -\operatorname {arctanh}\left (\cos \left (y\right )\right ) = x \] Verified OK.

\[ \operatorname {arctanh}\left (\cos \left (y\right )\right ) = x \] Verified OK.

\[ {}y^{\prime \prime } = 1+{y^{\prime }}^{2} \]

\[ -\arctan \left (\frac {1}{\sqrt {-1+{\mathrm e}^{2 y} c_{1}^{2}}}\right ) = x +c_{2} \] Verified OK.

\[ \arctan \left (\frac {1}{\sqrt {-1+{\mathrm e}^{2 y} c_{1}^{2}}}\right ) = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime } = \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} \]

\[ -\sqrt {-\left (y+c_{1} +1\right ) \left (y+c_{1} -1\right )} = x +c_{2} \] Verified OK.

\[ \sqrt {-y^{2}-2 y c_{1} -c_{1}^{2}+1} = x +c_{3} \] Verified OK.

\[ {}y y^{\prime \prime } = {y^{\prime }}^{2} \left (1-y^{\prime } \sin \left (y\right )-y y^{\prime } \cos \left (y\right )\right ) \]

\[ \int _{}^{y}\frac {\sin \left (\textit {\_a} \right ) \textit {\_a} -c_{2}}{\textit {\_a}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}\left (1+y^{2}\right ) y^{\prime \prime }+{y^{\prime }}^{3}+y^{\prime } = 0 \]

\[ \int _{}^{y}-\frac {1}{\tan \left (\arctan \left (\textit {\_a} \right )+c_{1} \right )}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}\left (y y^{\prime \prime }+1+{y^{\prime }}^{2}\right )^{2} = \left (1+{y^{\prime }}^{2}\right )^{3} \]

\[ \frac {y \left (c_{2} y-2\right )}{\sqrt {-c_{2} y \left (c_{2} y-2\right )}} = x +c_{5} \] Verified OK.

\[ \frac {\left (-c_{2} y+2\right ) y}{\sqrt {-c_{2} y \left (c_{2} y-2\right )}} = x +c_{6} \] Verified OK.

\[ -\sqrt {-y \left (y-2 c_{4} \right )} = x +c_{7} \] Verified OK.

\[ \sqrt {y \left (-y+2 c_{4} \right )} = x +c_{8} \] Verified OK.

\[ {}x y^{\prime \prime } = y^{\prime } \left (2-3 x y^{\prime }\right ) \]

\[ \frac {{\mathrm e}^{3 y}}{3}-\frac {c_{2} x^{3}}{3}-c_{3} = 0 \] Verified OK.

\[ {}5 {y^{\prime }}^{2}+6 x y^{\prime }-2 y = 0 \]

\[ y = 0 \] Verified OK.

\[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+10 y}}{5}+\frac {125 c_{1}}{\left (-15 x +5 \sqrt {9 x^{2}+10 y}\right )^{\frac {3}{2}}} \] Verified OK.

\[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+10 y}}{5}+\frac {125 c_{1}}{\left (-15 x -5 \sqrt {9 x^{2}+10 y}\right )^{\frac {3}{2}}} \] Verified OK.

\[ {}{y^{\prime }}^{4}+x y^{\prime }-3 y = 0 \]

\[ y = 0 \] Verified OK.

\[ x = \frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+x \textit {\_Z} -3 y\right )^{3}}{5}+c_{1} \sqrt {\operatorname {RootOf}\left (\textit {\_Z}^{4}+x \textit {\_Z} -3 y\right )} \] Verified OK.

\[ {}{y^{\prime }}^{3}-2 x y^{\prime }-y = 0 \]

\[ y = 0 \] Verified OK.

\[ x = \frac {{\left (\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{96 \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {c_{1} 6^{\frac {2}{3}}}{{\left (\frac {\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x}{\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}} \] Verified OK.

\[ x = \frac {{\left (-i \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+24 i \sqrt {3}\, x +\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{384 \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {c_{1} 12^{\frac {2}{3}}}{{\left (\frac {i \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}-24 i \sqrt {3}\, x -\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x}{\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}} \] Verified OK.

\[ x = \frac {{\left (i \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}-24 i \sqrt {3}\, x +\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{384 \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {2 c_{1} 18^{\frac {1}{3}}}{{\left (\frac {-i \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+24 i \sqrt {3}\, x -\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x}{\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}} \] Warning, solution could not be verified

\[ {}y {y^{\prime }}^{2}-\left (x +y\right ) y^{\prime }+y = 0 \]

\[ y = 0 \] Verified OK.

\[ y = x \] Verified OK.

\[ x = \frac {2 c_{2} x \,{\mathrm e}^{\frac {2 y}{x +y+\sqrt {\left (x +3 y\right ) \left (x -y\right )}}}}{x +y+\sqrt {\left (x +3 y\right ) \left (x -y\right )}} \] Verified OK.

\[ x = \frac {2 c_{2} x \,{\mathrm e}^{\frac {2 y}{x +y-\sqrt {\left (x +3 y\right ) \left (x -y\right )}}}}{x +y-\sqrt {\left (x +3 y\right ) \left (x -y\right )}} \] Verified OK.

\[ {}y^{\prime } = \frac {2 x -y}{x +4 y} \]

\[ -\left (x +y\right ) \left (x -2 y\right ) = 2 \] Verified OK.

\[ {}y^{2}+\cos \left (x \right )+\left (2 y x +\sin \left (y\right )\right ) y^{\prime } = 0 \]

\[ y^{2} x +\sin \left (x \right )-\cos \left (y\right ) = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {\cos \left (y\right ) \sec \left (x \right )}{x} \]

\[ \int _{}^{x}-\frac {\sec \left (\textit {\_a} \right )}{\textit {\_a}}d \textit {\_a} +\ln \left (\sec \left (y\right )+\tan \left (y\right )\right ) = c_{1} \] Verified OK.

\[ {}y^{\prime } = x \left (\cos \left (y\right )+y\right ) \]

\[ -\frac {x^{2}}{2}+\int _{0}^{y}\frac {1}{\cos \left (\textit {\_a} \right )+\textit {\_a}}d \textit {\_a} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {\sec \left (x \right ) \left (\sin \left (y\right )+y\right )}{x} \]

\[ \int _{}^{x}-\frac {\sec \left (\textit {\_a} \right )}{\textit {\_a}}d \textit {\_a} +\int _{0}^{y}\frac {1}{\sin \left (\textit {\_a} \right )+\textit {\_a}}d \textit {\_a} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \left (5+\frac {\sec \left (x \right )}{x}\right ) \left (\sin \left (y\right )+y\right ) \]

\[ \int _{}^{x}-\frac {\sec \left (\textit {\_a} \right )+5 \textit {\_a}}{\textit {\_a}}d \textit {\_a} +\int _{0}^{y}\frac {1}{\sin \left (\textit {\_a} \right )+\textit {\_a}}d \textit {\_a} = c_{1} \] Verified OK.

\[ {}y^{\prime } = x +\frac {\sec \left (x \right ) y}{x} \]

\[ \int _{}^{x}-\frac {\left (\sec \left (\textit {\_a} \right ) y+\textit {\_a}^{2}\right ) {\mathrm e}^{-\left (\int \frac {\sec \left (\textit {\_a} \right )}{\textit {\_a}}d \textit {\_a} \right )}}{\textit {\_a}}d \textit {\_a} +\left (-{\mathrm e}^{-\left (\int _{}^{x}\frac {\sec \left (\textit {\_a} \right )}{\textit {\_a}}d \textit {\_a} \right )}+{\mathrm e}^{-\left (\int \frac {\sec \left (x \right )}{x}d x \right )}\right ) y = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {\ln \left (1+y^{2}\right )}{\ln \left (x^{2}+1\right )} \]

\[ \int _{}^{x}-\frac {1}{\ln \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +\int _{0}^{y}\frac {1}{\ln \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {-y x -1}{4 x^{3} y-2 x^{2}} \]

\[ \frac {\left (4 y x +1\right )^{\frac {3}{5}} \left (y x -1\right )^{\frac {2}{5}} 2^{\frac {4}{5}}}{4 x} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \sqrt {\frac {y+1}{y^{2}}} \]

\[ \frac {2 y \left (y-2\right ) \sqrt {\frac {1+y}{y^{2}}}}{3} = x -\frac {2 \sqrt {2}}{3} \] Verified OK.

\[ {}y^{\prime } = \sqrt {y}+x \]

\[ \frac {\ln \left (x +2 \sqrt {y}\right )}{6}-\frac {\ln \left (-x +2 \sqrt {y}\right )}{6}-\frac {\ln \left (\sqrt {y}+x \right )}{3}+\frac {\ln \left (\sqrt {y}-x \right )}{3}+\frac {\ln \left (-x^{2}+y\right )}{3}+\frac {\ln \left (-x^{2}+4 y\right )}{6} = c_{1} \] Verified OK.

\[ {}2 t +3 x+\left (x+2\right ) x^{\prime } = 0 \]

\[ -\ln \left (x-1+t \right )+2 \ln \left (x-4+2 t \right ) = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {1}{1-y} \]

\[ -\frac {y \left (y-2\right )}{2} = t \] Verified OK.

\[ {}p^{\prime } = a p-b p^{2} \]

\[ \frac {\ln \left (p\right )-\ln \left (b p-a \right )}{a} = \frac {-\ln \left (b \operatorname {p0} -a \right )+\ln \left (\operatorname {p0} \right )+\left (t -\operatorname {t0} \right ) a}{a} \] Verified OK.

\[ {}y^{2}+\frac {2}{x}+2 x y y^{\prime } = 0 \]

\[ y^{2} x +2 \ln \left (x \right ) = c_{1} \] Verified OK.

\[ {}y y^{\prime } = 1-x {y^{\prime }}^{3} \]

\[ y = -x -1 \] Verified OK.

\[ x = \frac {54 \,2^{\frac {2}{3}} x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 3^{\frac {1}{3}} \left (\frac {2^{\frac {1}{3}} x \left (\sqrt {\frac {4 y^{3}+27 x}{x}}\, c_{1} 3^{\frac {1}{6}}-2 \left (y-\frac {3 c_{1}}{2}\right ) 3^{\frac {2}{3}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{3}+\frac {2^{\frac {2}{3}} 3^{\frac {5}{6}} x^{2} \sqrt {\frac {4 y^{3}+27 x}{x}}}{3}+3 x \left (\frac {2 y^{2} c_{1}}{9}+x \right ) 3^{\frac {1}{3}} 2^{\frac {2}{3}}+\left (-\frac {4 y c_{1}}{3}+x \right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )}{\left (2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}-2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} y x +6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2} \left (-2^{\frac {2}{3}} 3^{\frac {1}{3}} x y+{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2}} \] Warning, solution could not be verified

\[ x = -\frac {36 \,2^{\frac {2}{3}} x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\left (-\frac {8 y c_{1}}{9}+\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+\left (-\frac {\left (c_{1} \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) \sqrt {\frac {4 y^{3}+27 x}{x}}-6 \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) \left (y-\frac {3 c_{1}}{2}\right )\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+2^{\frac {2}{3}} \left (\frac {x \left (i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {4 y^{3}+27 x}{x}}}{3}+\left (\frac {2 y^{2} c_{1}}{9}+x \right ) \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right )\right )\right ) x \right ) 3^{\frac {1}{3}}}{{\left (\left (-i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+y 2^{\frac {2}{3}} x \left (i 3^{\frac {1}{3}}-3^{\frac {5}{6}}\right )\right )}^{2} \left (\frac {2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) y 2^{\frac {1}{3}}\right )\right )^{2}} \] Warning, solution could not be verified

\[ x = \frac {36 \,2^{\frac {2}{3}} \left (\left (\frac {8 y c_{1}}{9}-\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {2^{\frac {1}{3}} \left (c_{1} \left (i 3^{\frac {2}{3}}-3^{\frac {1}{6}}\right ) \sqrt {\frac {4 y^{3}+27 x}{x}}-6 \left (y-\frac {3 c_{1}}{2}\right ) \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {\left (i 3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{3}\right ) x \sqrt {\frac {4 y^{3}+27 x}{x}}}{3}+\left (\frac {2 y^{2} c_{1}}{9}+x \right ) \left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right )\right ) 2^{\frac {2}{3}}\right )\right ) x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 3^{\frac {1}{3}}}{\left (\frac {2^{\frac {2}{3}} \left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+y 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right )\right )\right )^{2} {\left (\left (-i+\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+y 2^{\frac {2}{3}} x \left (3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right )\right )}^{2}} \] Warning, solution could not be verified

\[ {}y y^{\prime \prime } = 1 \]

\[ \int _{}^{y}\frac {1}{\sqrt {2 \ln \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {1}{\sqrt {2 \ln \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}3 y y^{\prime \prime }+y = 5 \]

\[ \int _{}^{y}\frac {3}{\sqrt {-6 \textit {\_a} +30 \ln \left (\textit {\_a} \right )+18 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {3}{\sqrt {-6 \textit {\_a} +30 \ln \left (\textit {\_a} \right )+18 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}a y y^{\prime \prime }+b y = c \]

\[ \int _{}^{y}\frac {a}{\sqrt {-2 a \left (b \textit {\_a} -c \ln \left (\textit {\_a} \right )-c_{1} a \right )}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {a}{\sqrt {-2 a \left (b \textit {\_a} -c \ln \left (\textit {\_a} \right )-c_{1} a \right )}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}a y^{2} y^{\prime \prime }+b y^{2} = c \]

\[ \int _{}^{y}\frac {a \textit {\_a}}{\sqrt {-2 a \textit {\_a} \left (-c_{1} \textit {\_a} a +\textit {\_a}^{2} b +c \right )}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {a \textit {\_a}}{\sqrt {-2 a \textit {\_a} \left (-c_{1} \textit {\_a} a +\textit {\_a}^{2} b +c \right )}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}x^{\prime } = 4 A k \left (\frac {x}{A}\right )^{\frac {3}{4}}-3 k x \]

\[ -\frac {4 \ln \left (3 \left (\frac {x}{A}\right )^{\frac {1}{4}}-4\right )}{3 k} = t +c_{1} \] Verified OK.

\[ {}y^{\prime } = 2 y \left (x \sqrt {y}-1\right ) \]

\[ \frac {1}{\sqrt {y}} = x +1 \] Verified OK.

\[ {}w^{\prime } = -\frac {1}{2}-\frac {\sqrt {1-12 w}}{2} \]

\[ \frac {\sqrt {1-12 w}}{3}-\frac {\ln \left (1+\sqrt {1-12 w}\right )}{3} = z -1+\frac {\sqrt {13}}{3}-\frac {\ln \left (1+\sqrt {13}\right )}{3} \] Verified OK.

\[ {}y^{\prime \prime }+\sin \left (y\right ) {y^{\prime }}^{2} = 0 \]

\[ \int _{}^{y}\frac {{\mathrm e}^{-\cos \left (\textit {\_a} \right )}}{c_{1}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}v v^{\prime } = \frac {2 v^{2}}{r^{3}}+\frac {\lambda r}{3} \]

\[ -\frac {\lambda \,\operatorname {expIntegral}_{1}\left (-\frac {2}{r^{2}}\right )}{3}-\frac {{\mathrm e}^{\frac {2}{r^{2}}} \left (\lambda \,r^{2}-3 v^{2}\right )}{6} = c_{1} \] Verified OK.

\[ {}\left (y-2 x y^{\prime }\right )^{2} = {y^{\prime }}^{3} \]

\[ y = 0 \] Verified OK.

\[ x = \frac {\frac {432 \left (\left (16 \left (x^{3}-\frac {3 y}{16}\right ) \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}-128 x^{6}+160 y x^{3}-27 y^{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}-2048 \left (-\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}}{4}+\left (-\frac {\sqrt {3}\, \left (x^{3}-\frac {15 y}{16}\right ) \sqrt {-32 y x^{3}+27 y^{2}}}{4}+x^{6}-\frac {7 y x^{3}}{2}+\frac {135 y^{2}}{64}\right ) x \right ) x \right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}-4 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}}}}{5}-82944 c_{1} \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}} \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}}{16}-\frac {27 y}{16}\right )}{{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}-4 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \] Warning, solution could not be verified

\[ x = \frac {\left (\left (82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 y x^{3}+27 y^{2}}-221184 \left (x^{6}-\frac {5 y x^{3}}{4}+\frac {27 y^{2}}{128}\right ) \left (i \sqrt {3}-1\right )\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}+3538944 x \left (\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (-\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (i+\frac {\sqrt {3}}{3}\right ) \sqrt {-32 y x^{3}+27 y^{2}}}{4}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {7 y x^{3}}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2} {\left (-i \left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right ) \sqrt {3}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}}}-6635520 c_{1} \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}} \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}}{16}-\frac {27 y}{16}\right )}{5 \left (16 i \sqrt {3}\, x^{2}-i \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+16 x^{2}+8 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \] Warning, solution could not be verified

\[ x = \frac {\left (\left (-82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (i+\frac {\sqrt {3}}{3}\right ) \sqrt {-32 y x^{3}+27 y^{2}}+221184 \left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {5 y x^{3}}{4}+\frac {27 y^{2}}{128}\right )\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}-3538944 \left (-\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (-\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 y x^{3}+27 y^{2}}}{4}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {7 y x^{3}}{2}+\frac {135 y^{2}}{64}\right )\right )\right ) x \right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2} {\left (i \left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right ) \sqrt {3}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}}}-6635520 c_{1} \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}} \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}}{16}-\frac {27 y}{16}\right )}{5 \left (16 i \sqrt {3}\, x^{2}-i \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}-16 x^{2}-8 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}-\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \] Warning, solution could not be verified

\[ y = 0 \] Verified OK.

\[ x = \frac {\frac {432 \left (\left (16 \left (x^{3}-\frac {3 y}{16}\right ) \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}+128 x^{6}-160 y x^{3}+27 y^{2}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}+2048 \left (\frac {\left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}}{4}+\left (\frac {\sqrt {3}\, \left (x^{3}-\frac {15 y}{16}\right ) \sqrt {-32 y x^{3}+27 y^{2}}}{4}+x^{6}-\frac {7 y x^{3}}{2}+\frac {135 y^{2}}{64}\right ) x \right ) x \right ) \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}+4 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}}}}{5}+82944 c_{2} \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}}{{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}+4 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \] Warning, solution could not be verified

\[ x = \frac {\left (\left (82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 y x^{3}+27 y^{2}}+221184 \left (x^{6}-\frac {5 y x^{3}}{4}+\frac {27 y^{2}}{128}\right ) \left (i \sqrt {3}-1\right )\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}-3538944 \left (-\frac {\left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (i+\frac {\sqrt {3}}{3}\right ) \sqrt {-32 y x^{3}+27 y^{2}}}{4}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {7 y x^{3}}{2}+\frac {135 y^{2}}{64}\right )\right )\right ) x \right ) \sqrt {\frac {{\left (-i \left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}-16 x^{2}\right ) \sqrt {3}+{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2}\right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}}}+6635520 c_{2} \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}}{5 \left (16 i \sqrt {3}\, x^{2}-i \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+16 x^{2}-8 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \] Warning, solution could not be verified

\[ x = \frac {\left (\left (-82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (i+\frac {\sqrt {3}}{3}\right ) \sqrt {-32 y x^{3}+27 y^{2}}-221184 \left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {5 y x^{3}}{4}+\frac {27 y^{2}}{128}\right )\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}+3538944 \left (\frac {\left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 y x^{3}+27 y^{2}}}{4}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {7 y x^{3}}{2}+\frac {135 y^{2}}{64}\right )\right )\right ) x \right ) \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2} {\left (i \left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right ) \sqrt {3}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}}}+6635520 c_{2} \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}}{5 \left (16 i \sqrt {3}\, x^{2}-i \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}-16 x^{2}+8 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {1}{3}}-\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 y x^{3}+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \] Warning, solution could not be verified

\[ {}y^{\prime \prime } = A y^{\frac {2}{3}} \]

\[ \int _{}^{y}\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y^{\prime } = \frac {y}{2 y \ln \left (y\right )+y-x} \]

\[ y^{2} \ln \left (y\right )-y x = c_{1} \] Verified OK.

\[ {}c y^{\prime } = \frac {x a +b y^{2}}{y} \]

\[ \frac {c \left (2 y^{2} b^{2}+2 a b x +a c \right ) {\mathrm e}^{-\frac {2 b x}{c}}}{4 b^{2}} = c_{1} \] Verified OK.

\[ {}{y^{\prime }}^{2} = \frac {y^{2}}{x} \]

\[ \frac {y \ln \left (y\right )}{\sqrt {y^{2}}} = 2 x \sqrt {\frac {1}{x}}+c_{1} \] Verified OK. {0 < 1/x, 0 < y^2}

\[ -\frac {y \ln \left (y\right )}{\sqrt {y^{2}}} = 2 x \sqrt {\frac {1}{x}}+c_{1} \] Verified OK. {0 < 1/x, 0 < y^2}

\[ {}y^{\prime } = \left (1+6 x +y\right )^{\frac {1}{4}} \]

\[ x = -216 \ln \left (-y-6 x +1295\right )-12 \sqrt {1+6 x +y}+216 \ln \left (\sqrt {1+6 x +y}+36\right )-216 \ln \left (-36+\sqrt {1+6 x +y}\right )+144 \left (1+6 x +y\right )^{\frac {1}{4}}+432 \ln \left (\left (1+6 x +y\right )^{\frac {1}{4}}-6\right )-432 \ln \left (\left (1+6 x +y\right )^{\frac {1}{4}}+6\right )+\frac {4 \left (1+6 x +y\right )^{\frac {3}{4}}}{3}+c_{1} \] Verified OK.

\[ {}y^{\prime } = \left (a +b x +y\right )^{4} \]

\[ x = \int _{}^{y}\frac {1}{\left (b x +\textit {\_a} \right )^{4}+4 \left (b x +\textit {\_a} \right )^{3} a +6 \left (b x +\textit {\_a} \right )^{2} a^{2}+4 \left (b x +\textit {\_a} \right ) a^{3}+a^{4}+b}d \textit {\_a} +c_{1} \] Verified OK.

\[ {}y^{\prime } = \left (\pi +x +7 y\right )^{\frac {7}{2}} \]

\[ x -7 \left (\int _{}^{y}\frac {1}{7 \left (\pi +x +7 \textit {\_a} \right )^{\frac {7}{2}}+1}d \textit {\_a} \right )-c_{1} = 0 \] Verified OK.

\[ {}{y^{\prime \prime }}^{2}+y^{\prime } = 1 \]

\[ \int _{}^{y}-\frac {4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}}{\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}}+4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ \int _{}^{y}\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}+16+\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}}-8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = x +c_{4} \] Verified OK.

\[ \int _{}^{y}-\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}-\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}}-16+8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = x +c_{5} \] Verified OK.

\[ \int _{}^{y}-\frac {4 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 c_{2}^{2}+18 c_{2} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}}{\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 c_{2}^{2}+18 c_{2} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}}+4 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 c_{2}^{2}+18 c_{2} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{6} \] Verified OK.

\[ \int _{}^{y}\frac {8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 c_{2}^{2}+18 c_{2} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}}{i \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 c_{2}^{2}+18 c_{2} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}+\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 c_{2}^{2}+18 c_{2} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}}+16-8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 c_{2}^{2}+18 c_{2} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = x +c_{7} \] Verified OK.

\[ \int _{}^{y}-\frac {8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 c_{2}^{2}+18 c_{2} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}}{i \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 c_{2}^{2}+18 c_{2} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}-\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 c_{2}^{2}+18 c_{2} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}}-16+8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 c_{2}^{2}+18 c_{2} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = x +c_{8} \] Verified OK.

\[ {}y^{\prime \prime }+{y^{\prime }}^{2} = 1 \]

\[ \int _{}^{y}\frac {1}{\sqrt {1+\frac {{\mathrm e}^{-2 \textit {\_a}}}{c_{1}^{2}}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ -\frac {{\mathrm e}^{-y} \sqrt {\frac {{\mathrm e}^{2 y} c_{1}^{2}+1}{c_{1}^{2}}}\, \ln \left ({\mathrm e}^{y}+\sqrt {\frac {{\mathrm e}^{2 y} c_{1}^{2}+1}{c_{1}^{2}}}\right )}{\sqrt {\frac {\left ({\mathrm e}^{2 y} c_{1}^{2}+1\right ) {\mathrm e}^{-2 y}}{c_{1}^{2}}}} = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime }+{y^{\prime }}^{2}+y = 0 \]

\[ \int _{}^{y}\frac {2 \,{\mathrm e}^{2 \textit {\_a}}}{\sqrt {-2 \,{\mathrm e}^{2 \textit {\_a}} \left (2 \,{\mathrm e}^{2 \textit {\_a}} \textit {\_a} -{\mathrm e}^{2 \textit {\_a}}-4 c_{1} \right )}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {2 \,{\mathrm e}^{2 \textit {\_a}}}{\sqrt {-2 \,{\mathrm e}^{2 \textit {\_a}} \left (2 \,{\mathrm e}^{2 \textit {\_a}} \textit {\_a} -{\mathrm e}^{2 \textit {\_a}}-4 c_{1} \right )}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y {y^{\prime \prime }}^{2}+y^{\prime } = 0 \]

\[ \int _{}^{y}\frac {4}{\left (\frac {\left (12 \textit {\_a} -4\right ) \left (\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ \int _{}^{y}\frac {16}{\left (\frac {\left (12 \textit {\_a} -4\right ) \left (\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{4} \] Verified OK.

\[ \int _{}^{y}\frac {16}{\left (\frac {\left (12 \textit {\_a} -4\right ) \left (\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{5} \] Verified OK.

\[ \frac {3 y^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], -\frac {y \left (12 \sqrt {-\frac {1}{y}}\, \sqrt {y}+12 i\right )}{4 \sqrt {-\frac {1}{y}}\, \sqrt {y}+4 i}\right )}{\left (4 \sqrt {-\frac {1}{y}}\, \sqrt {y}+4 i\right )^{\frac {2}{3}}} = x +c_{6} \] Verified OK.

\[ \frac {12 y^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], -\frac {y \left (12 \sqrt {-\frac {1}{y}}\, \sqrt {y}+12 i\right )}{4 \sqrt {-\frac {1}{y}}\, \sqrt {y}+4 i}\right )}{\left (4 \sqrt {-\frac {1}{y}}\, \sqrt {y}+4 i\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}} = x +c_{7} \] Verified OK.

\[ \frac {12 y^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], -\frac {y \left (12 \sqrt {-\frac {1}{y}}\, \sqrt {y}+12 i\right )}{4 \sqrt {-\frac {1}{y}}\, \sqrt {y}+4 i}\right )}{\left (4 \sqrt {-\frac {1}{y}}\, \sqrt {y}+4 i\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}} = x +c_{8} \] Verified OK.

\[ \int _{}^{y}\frac {4}{\left (\frac {\left (-12 \textit {\_a} -4\right ) \left (-\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{9} \] Verified OK.

\[ \int _{}^{y}\frac {16}{\left (\frac {\left (-12 \textit {\_a} -4\right ) \left (-\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{10} \] Verified OK.

\[ \int _{}^{y}\frac {16}{\left (\frac {\left (-12 \textit {\_a} -4\right ) \left (-\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{11} \] Verified OK.

\[ \int _{}^{y}\frac {4}{\left (\frac {\left (-12 \textit {\_a} +4\right ) \left (-\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{12} \] Verified OK.

\[ \int _{}^{y}\frac {16}{\left (\frac {\left (-12 \textit {\_a} +4\right ) \left (-\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{13} \] Verified OK.

\[ \int _{}^{y}\frac {16}{\left (\frac {\left (-12 \textit {\_a} +4\right ) \left (-\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{14} \] Verified OK.

\[ {}y {y^{\prime \prime }}^{2}+{y^{\prime }}^{3} = 0 \]

\[ -\frac {2 c_{1}^{2}}{c_{1}^{2}+4 y}-\ln \left (c_{1}^{2}+4 y\right )-\frac {2 \sqrt {-\frac {1}{y}}\, \sqrt {y}\, \left (\arctan \left (\frac {2 \sqrt {y}}{c_{1}}\right ) c_{1}^{2}+4 \arctan \left (\frac {2 \sqrt {y}}{c_{1}}\right ) y-2 \sqrt {y}\, c_{1} \right )}{c_{1}^{2}+4 y} = x +c_{2} \] Verified OK.

\[ {}y^{2} {y^{\prime \prime }}^{2}+y^{\prime } = 0 \]

\[ \int _{}^{y}-\frac {4}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ \int _{}^{y}-\frac {16}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{4} \] Verified OK.

\[ \int _{}^{y}-\frac {16}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{5} \] Verified OK.

\[ \int _{}^{y}-\frac {4}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{6} \] Verified OK.

\[ \int _{}^{y}-\frac {16}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{7} \] Verified OK.

\[ \int _{}^{y}-\frac {16}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{8} \] Verified OK.

\[ {}y {y^{\prime \prime }}^{4}+{y^{\prime }}^{2} = 0 \]

\[ \int _{}^{y}\frac {\textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}-\left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{5} \] Verified OK.

\[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}-\left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{6} \] Verified OK.

\[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}-\left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{7} \] Verified OK.

\[ \int _{}^{y}\frac {\textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}+i \left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{8} \] Verified OK.

\[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}+i \left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{9} \] Verified OK.

\[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}+i \left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{10} \] Verified OK.

\[ \int _{}^{y}\frac {\textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}+\left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{11} \] Verified OK.

\[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}+\left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{12} \] Verified OK.

\[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}+\left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{13} \] Verified OK.

\[ \int _{}^{y}\frac {\textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}-i \left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{14} \] Verified OK.

\[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}-i \left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{15} \] Verified OK.

\[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}-i \left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{16} \] Verified OK.

\[ {}y^{3} {y^{\prime \prime }}^{2}+y y^{\prime } = 0 \]

\[ \int _{}^{y}-\frac {4}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ \int _{}^{y}-\frac {16}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{4} \] Verified OK.

\[ \int _{}^{y}-\frac {16}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{5} \] Verified OK.

\[ \int _{}^{y}-\frac {4}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{6} \] Verified OK.

\[ \int _{}^{y}-\frac {16}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{7} \] Verified OK.

\[ \int _{}^{y}-\frac {16}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{8} \] Verified OK.

\[ {}y {y^{\prime \prime }}^{3}+y^{3} {y^{\prime }}^{5} = 0 \]

\[ \int _{}^{y}\frac {1}{-\frac {\textit {\_a}^{5}}{125}+\frac {\textit {\_a}^{2} \left (-\textit {\_a}^{2}\right )^{\frac {2}{3}} c_{1}}{25}+\frac {\textit {\_a} \left (-\textit {\_a}^{2}\right )^{\frac {1}{3}} c_{1}^{2}}{15}+\frac {c_{1}^{3}}{27}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime \prime }+x y^{\prime }+y {y^{\prime }}^{2} = 0 \]

\[ \frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {2}\, y}{2}\right )}{2}-\frac {c_{2} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )}{2}-c_{3} = 0 \] Verified OK.

\[ {}y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+y {y^{\prime }}^{2} = 0 \]

\[ \frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {2}\, y}{2}\right )}{2}-\left (\int c_{2} {\mathrm e}^{\cos \left (x \right )}d x \right )-c_{3} = 0 \] Verified OK.

\[ {}y^{\prime \prime } y^{\prime }+y^{2} = 0 \]

\[ \int _{}^{y}\frac {1}{\left (-\textit {\_a}^{3}+3 c_{1} \right )^{\frac {1}{3}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}\frac {1}{-\frac {\left (-\textit {\_a}^{3}+3 c_{1} \right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (-\textit {\_a}^{3}+3 c_{1} \right )^{\frac {1}{3}}}{2}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ \int _{}^{y}\frac {1}{-\frac {\left (-\textit {\_a}^{3}+3 c_{1} \right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (-\textit {\_a}^{3}+3 c_{1} \right )^{\frac {1}{3}}}{2}}d \textit {\_a} = x +c_{4} \] Verified OK.

\[ {}y^{\prime \prime } y^{\prime }+y^{n} = 0 \]

\[ \int _{}^{y}\frac {n +1}{{\left (\left (3 c_{1} n -3 \textit {\_a}^{n +1}+3 c_{1} \right ) \left (n +1\right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {2 \left (n +1\right )}{{\left (\left (3 c_{1} n -3 \textit {\_a}^{n +1}+3 c_{1} \right ) \left (n +1\right )^{2}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ \int _{}^{y}\frac {2 n +2}{{\left (\left (3 c_{1} n -3 \textit {\_a}^{n +1}+3 c_{1} \right ) \left (n +1\right )^{2}\right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}d \textit {\_a} = x +c_{4} \] Verified OK.

\[ {}y^{\prime \prime }+x y^{\prime }+y {y^{\prime }}^{2} = 0 \]

\[ \frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {2}\, y}{2}\right )}{2}-\frac {c_{2} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )}{2}-c_{3} = 0 \] Verified OK.

\[ {}y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+{y^{\prime }}^{2} = 0 \]

\[ {\mathrm e}^{y}-\left (\int c_{2} {\mathrm e}^{\cos \left (x \right )}d x \right )-c_{3} = 0 \] Verified OK.

\[ {}3 y^{\prime \prime }+\cos \left (x \right ) y^{\prime }+\sin \left (y\right ) {y^{\prime }}^{2} = 0 \]

\[ \int _{}^{y}{\mathrm e}^{-\frac {\cos \left (\textit {\_a} \right )}{3}}d \textit {\_a} -\left (\int c_{2} {\mathrm e}^{-\frac {\sin \left (x \right )}{3}}d x \right )-c_{3} = 0 \] Verified OK.

\[ {}10 y^{\prime \prime }+y^{\prime } x^{2}+\frac {3 {y^{\prime }}^{2}}{y} = 0 \]

\[ \frac {10 y^{\frac {13}{10}}}{13}-\frac {10^{\frac {1}{3}} 9^{\frac {2}{3}} c_{2} \left (\frac {3 \,243^{\frac {1}{6}} 10^{\frac {5}{6}} x \,{\mathrm e}^{-\frac {x^{3}}{60}} \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {x^{3}}{30}\right )}{40 \left (x^{3}\right )^{\frac {1}{6}}}+\frac {3 \,30^{\frac {5}{6}} {\mathrm e}^{-\frac {x^{3}}{60}} \operatorname {WhittakerM}\left (\frac {7}{6}, \frac {2}{3}, \frac {x^{3}}{30}\right )}{x^{2} \left (x^{3}\right )^{\frac {1}{6}}}\right )}{9}-c_{3} = 0 \] Verified OK.

\[ {}2 x y^{2}-y+\left (y^{2}+x +y\right ) y^{\prime } = 0 \]

\[ \frac {x \left (x y-1\right )}{y}+y+\ln \left (y\right ) = c_{1} \] Verified OK.

\[ {}y^{\prime }+y \cos \left (x \right )-{\mathrm e}^{2 x} = 0 \]

\[ \int _{}^{x}\left (y \cos \left (\textit {\_a} \right )-{\mathrm e}^{2 \textit {\_a}}\right ) {\mathrm e}^{\sin \left (\textit {\_a} \right )}d \textit {\_a} = c_{1} \] Verified OK.