\[ {}5 t +2 y+1+\left (2 t +y+1\right ) y^{\prime } = 0 \]

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

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

\[ \frac {1}{y^{3}} = \frac {\sin \left (x \right )^{3} \left (3 \cot \left (x \right ) \csc \left (x \right )-3 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+2 c_{1} \right )}{2} \] Warning, solution could not be verified

\[ {}y = -t y^{\prime }+\frac {{y^{\prime }}^{5}}{5} \]

\[ y = 0 \] Verified OK.

\[ t = \frac {\operatorname {RootOf}\left (\textit {\_Z}^{5}-5 t \textit {\_Z} -5 y\right )^{\frac {9}{2}}+9 c_{1}}{9 \sqrt {\operatorname {RootOf}\left (\textit {\_Z}^{5}-5 t \textit {\_Z} -5 y\right )}} \] Verified OK.

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

\[ y = 0 \] Verified OK.

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

\[ t = \frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {1}{3}}}{3}+\frac {\left (t +3\right )^{2}}{3 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {1}{3}}}+\frac {t}{3}-1+\frac {36 c_{1} \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {2}{3}}}{{\left (\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {2}{3}}+\left (t -3\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {1}{3}}+\left (t +3\right )^{2}\right )}^{2}} \] Verified OK.

\[ t = -1-\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {1}{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {1}{3}}}+\frac {t}{3}-\frac {i \left (-\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {2}{3}}+\left (t +3\right )^{2}\right ) \sqrt {3}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {1}{3}}}+\frac {144 c_{1} \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {2}{3}}}{{\left (\left (i \sqrt {3}-1\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {2}{3}}+\left (2 t -6\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {1}{3}}-\left (1+i \sqrt {3}\right ) \left (t +3\right )^{2}\right )}^{2}} \] Verified OK.

\[ t = -\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {1}{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {1}{3}}}+\frac {t}{3}-1+\frac {i \left (-\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {2}{3}}+\left (t +3\right )^{2}\right ) \sqrt {3}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {1}{3}}}+\frac {144 c_{1} \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {2}{3}}}{{\left (\left (1+i \sqrt {3}\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {2}{3}}+\left (-2 t +6\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27+9 t^{2}+27 t -27 y+t^{3}\right )}\right )^{\frac {1}{3}}-\left (i \sqrt {3}-1\right ) \left (t +3\right )^{2}\right )}^{2}} \] Verified OK.

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

\[ y = t +3 \] Verified OK.

\[ t = \frac {t}{3}+\frac {\sqrt {t^{2}+8 y-16 t -8}}{3}+\frac {8}{3}+\frac {2 c_{1}}{\sqrt {t +\sqrt {t^{2}+8 y-16 t -8}-4}} \] Verified OK.

\[ t = \frac {t}{3}-\frac {\sqrt {t^{2}+8 y-16 t -8}}{3}+\frac {8}{3}+\frac {2 c_{1}}{\sqrt {t -\sqrt {t^{2}+8 y-16 t -8}-4}} \] Verified OK.

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

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

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

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

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

\[ \operatorname {Si}\left (\frac {t}{y}\right )-\ln \left (y\right ) = \operatorname {Si}\left (\frac {1}{2}\right )-\ln \left (2\right ) \] Verified OK.

\[ {}y^{\prime } = \frac {2 t^{5}}{5 y^{2}} \]

\[ -\frac {t^{6}}{6}+\frac {5 y^{3}}{6} = c_{1} \] Verified OK.

\[ {}\cos \left (4 x \right )-8 y^{\prime } \sin \left (y\right ) = 0 \]

\[ -\frac {\sin \left (4 x \right )}{4}-8 \cos \left (y\right ) = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {{\mathrm e}^{5 t}}{y^{4}} \]

\[ \frac {y^{5}}{5}-\frac {{\mathrm e}^{5 t}}{5} = c_{1} \] Verified OK.

\[ {}-\frac {1}{x^{5}}+\frac {1}{x^{3}} = \left (2 y^{4}-6 y^{9}\right ) y^{\prime } \]

\[ \frac {2 x^{2}-1}{4 x^{4}}-\frac {\left (3 y^{5}-1\right )^{2}}{15} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {y \,{\mathrm e}^{-2 t}}{\ln \left (y\right )} \]

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

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

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

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

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

\[ {}r^{\prime } = \frac {r^{2}+t^{2}}{r t} \]

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

\[ {}x^{\prime } = \frac {5 t x}{t^{2}+x^{2}} \]

\[ -\frac {5 t^{2}}{2 x^{\frac {2}{5}}}+\frac {5 x^{\frac {8}{5}}}{8} = c_{1} \] Verified OK.

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

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

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

\[ \frac {y t^{3}}{3}-\cos \left (t \right )-\sin \left (y\right ) = c_{1} \] Verified OK.

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

\[ -\frac {t \left (-2 \tan \left (y\right )+t \right )}{2}+y = c_{1} \] Verified OK.

\[ {}y^{\prime }+y = \frac {{\mathrm e}^{t}}{y^{2}} \]

\[ \frac {y^{3} {\mathrm e}^{3 t}}{3}-\frac {{\mathrm e}^{4 t}}{4} = c_{1} \] Verified OK.

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

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

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

\[ y+\sin \left (t -y\right ) = \pi \] Verified OK.

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

\[ -y \sin \left (t \right )+t \sin \left (y\right ) = 0 \] Verified OK.

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

\[ y^{2} t +\frac {\cos \left (2 y\right )}{2} = {\frac {1}{2}} \] Verified OK.

\[ {}y^{\prime } = \frac {t}{y^{3}} \]

\[ \frac {y^{4}}{4}-\frac {t^{2}}{2} = 0 \] Verified OK.

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

\[ \frac {y^{2}}{2}-t +\frac {1}{2} = 0 \] Verified OK.

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

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

\[ {}y^{\prime } = y+3 y^{\frac {1}{3}} \]

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

\[ {}y^{\prime } = 1-\cot \left (y\right ) \]

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

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

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

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

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

\[ {}1+y^{2}+x y y^{\prime } = 0 \]

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

\[ {}x \sqrt {1+y^{2}}+y y^{\prime } \sqrt {x^{2}+1} = 0 \]

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

\[ {}x \sqrt {1-y^{2}}+y \sqrt {-x^{2}+1}\, y^{\prime } = 0 \]

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

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

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

\[ {}{\mathrm e}^{x} \sin \left (y\right )^{3}+\left ({\mathrm e}^{2 x}+1\right ) \cos \left (y\right ) y^{\prime } = 0 \]

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

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

\[ \frac {1}{x}-\sin \left (y\right ) = \frac {\sqrt {3}}{2} \] Warning, solution could not be verified

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

\[ \frac {1}{x}-\frac {\cot \left (y\right )}{2} = -\frac {\sqrt {3}}{6} \] Warning, solution could not be verified

\[ {}x^{3} y^{\prime }-\sin \left (y\right ) = 1 \]

\[ \frac {1}{2 x^{2}}-\frac {2}{\tan \left (\frac {y}{2}\right )+1} = 0 \] Verified OK.

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

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

\[ {}{\mathrm e}^{y} = {\mathrm e}^{4 y} y^{\prime }+1 \]

\[ \int _{}^{y}\frac {{\mathrm e}^{4 \textit {\_a}}}{{\mathrm e}^{\textit {\_a}}-1}d \textit {\_a} = x +c_{1} \] Verified OK.

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

\[ \frac {1}{x}-\frac {1}{\tan \left (y\right )-1} = {\frac {1}{2}} \] Warning, solution could not be verified

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

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

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

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

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

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

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

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

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

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

\[ {}8 x +4 y+1+\left (4 x +2 y+1\right ) y^{\prime } = 0 \]

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

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

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

\[ {}4 y^{6}+x^{3} = 6 x y^{5} y^{\prime } \]

\[ \frac {-y^{6}-x^{3}}{x^{4}} = c_{1} \] Verified OK.

\[ {}y \left (1+\sqrt {x^{2} y^{4}+1}\right )+2 x y^{\prime } = 0 \]

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

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

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

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

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

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

\[ -{\mathrm e}^{\frac {y^{2}}{2}} x +\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, y}{2}\right )}{4} = c_{1} \] Verified OK.

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

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

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

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

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

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

\[ {}y^{\prime }-\tan \left (y\right ) = \frac {{\mathrm e}^{x}}{\cos \left (y\right )} \]

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

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

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

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

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

\[ {}3 x^{2}+6 x y^{2}+\left (6 x^{2} y+4 y^{3}\right ) y^{\prime } = 0 \]

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

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

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

\[ {}3 x^{2} \tan \left (y\right )-\frac {2 y^{3}}{x^{3}}+\left (x^{3} \sec \left (y\right )^{2}+4 y^{3}+\frac {3 y^{2}}{x^{2}}\right ) y^{\prime } = 0 \]

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

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

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

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

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

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

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

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

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

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

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

\[ {}\frac {2 x}{y^{3}}+\frac {\left (y^{2}-3 x^{2}\right ) y^{\prime }}{y^{4}} = 0 \]

\[ \frac {x^{2}}{y^{3}}-\frac {1}{y} = 0 \] Verified OK.

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

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

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

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

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

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

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

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

\[ {}2 x^{2} y+2 y+5+\left (2 x^{3}+2 x \right ) y^{\prime } = 0 \]

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

\[ {}x^{4} \ln \left (x \right )-2 x y^{3}+3 x^{2} y^{2} y^{\prime } = 0 \]

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

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

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

\[ {}2 x y^{2}-3 y^{3}+\left (7-3 x y^{2}\right ) y^{\prime } = 0 \]

\[ x \left (x -3 y\right )-\frac {7}{y} = c_{1} \] Verified OK.

\[ {}3 y^{2}-x +\left (2 y^{3}-6 y x \right ) y^{\prime } = 0 \]

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

\[ {}x^{2}+y^{2}+1-2 x y y^{\prime } = 0 \]

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

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

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

\[ {}y^{\frac {2}{5}}+{y^{\prime }}^{\frac {2}{5}} = a^{\frac {2}{5}} \]

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

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

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

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

\[ \int _{}^{y}\frac {1}{\sin \left (\operatorname {RootOf}\left (-\textit {\_a} +\textit {\_Z} +\ln \left (\sin \left (\textit {\_Z} \right )^{2}+1\right )\right )\right )}d \textit {\_a} = x +c_{1} \] Warning, solution could not be verified

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

\[ y = -\infty \] Warning, solution could not be verified

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

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

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

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

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

\[ y = 0 \] Verified OK.

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

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

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

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

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

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

\[ {}y = \frac {3 x y^{\prime }}{2}+{\mathrm e}^{y^{\prime }} \]

\[ y = 1 \] Verified OK.

\[ x = -\frac {27 \left (\left (-2 \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )^{2} x^{2}-4 \left (x -\frac {2 y}{3}\right ) x \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )-4 x^{2}+\frac {8 y x}{3}-\frac {8 y^{2}}{9}\right ) {\mathrm e}^{-\frac {3 x \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )-2 y}{3 x}}+c_{1} x^{2}\right ) x}{{\left (3 x \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )-2 y\right )}^{3}} \] Warning, solution could not be verified

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

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

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

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

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

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

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

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

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

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

\[ {}5 y x -4 y^{2}-6 x^{2}+\left (y^{2}-8 y x +\frac {5 x^{2}}{2}\right ) y^{\prime } = 0 \]

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

\[ {}3 x y^{2}-x^{2}+\left (3 x^{2} y-6 y^{2}-1\right ) y^{\prime } = 0 \]

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

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

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

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

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

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

\[ 4 \cos \left (\frac {x}{2}\right )-2 \ln \left (\csc \left (\frac {y}{2}\right )+\cot \left (\frac {y}{2}\right )\right ) = c_{1} \] Verified OK.

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

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

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

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