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

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

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

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

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

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

\[ {}t^{3} y^{2}+\frac {t^{4} y^{\prime }}{y^{6}} = 0 \]

\[ \frac {1}{7 y^{7}}-\ln \left (t \right ) = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

\[ \frac {y}{x}-\ln \left (\frac {y}{x}\right )-2 \ln \left (x \right )-c_{2} = 0 \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}\frac {r \tan \left (\theta \right ) r^{\prime }}{a^{2}-r^{2}} = 1 \]

\[ -\ln \left (\sin \left (\theta \right )\right )-\frac {\ln \left (r^{2}-a^{2}\right )}{2} = -\frac {\ln \left (-a^{2}\right )}{2}+\frac {\ln \left (2\right )}{2} \] Verified OK.

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

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

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

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

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

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

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

\[ \frac {\left (\frac {y^{3}}{x^{3}}-2\right ) x}{2 y}+\frac {1}{x^{2}}-\ln \left (x \right )-c_{2} = 0 \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}2 x +3 y+4+\left (3 x +4 y+5\right ) y^{\prime } = 0 \]

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

\[ {}2 u^{2}+2 u v+\left (u^{2}+v^{2}\right ) v^{\prime } = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y+x^{3} y+2 x^{2}+\left (x +4 y^{4} x +8 y^{3}\right ) y^{\prime } = 0 \]

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

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

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

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

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

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

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

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

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

\[ {}\left (2 s-{\mathrm e}^{2 t}\right ) s^{\prime } = 2 s \,{\mathrm e}^{2 t}-2 \cos \left (2 t \right ) \]

\[ -s \,{\mathrm e}^{2 t}+\sin \left (2 t \right )+s^{2} = c_{1} \] Verified OK.

\[ {}x y^{\prime }+y-x^{3} y^{6} = 0 \]

\[ \frac {5 x^{3} y^{5}-2}{10 x^{5} y^{5}} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}8 y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \]

\[ y = 0 \] Verified OK.

\[ y = -\frac {\sqrt {2}\, x}{4} \] Verified OK.

\[ y = \frac {\sqrt {2}\, x}{4} \] Verified OK.

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

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

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

\[ y = 0 \] Verified OK.

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

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

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

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

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

\[ y = 0 \] Verified OK.

\[ x = \frac {\left (-16 x^{2}-2 y\right ) \sqrt {4 x^{2}+2 y}+32 x^{3}+12 y x +3 c_{1}}{3 \left (2 x -\sqrt {4 x^{2}+2 y}\right )^{2}} \] Verified OK.

\[ x = \frac {2 \left (8 x^{2}+y\right ) \sqrt {4 x^{2}+2 y}+32 x^{3}+12 y x +3 c_{1}}{3 \left (2 x +\sqrt {4 x^{2}+2 y}\right )^{2}} \] Verified OK.

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

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

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

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

\[ \frac {y^{2}}{2}-\frac {x^{2}}{2}-\frac {c_{1} x}{2}-c_{2} = 0 \] Warning, solution could not be verified

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ \frac {y^{2} \left (y^{2}+1\right )}{2}+\frac {\left (2 x^{2}+1\right )^{2}}{8}-c_{1} = 0 \] Verified OK.

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

\[ 2 \sqrt {y}+2 \sqrt {x}-c_{1} = 0 \] Verified OK.

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

\[ -\sqrt {1-y^{2}}+x^{2}-c_{1} = 0 \] Verified OK.

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

\[ 2 \sqrt {y}-2 \sqrt {x}-c_{1} = 0 \] Verified OK.

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

\[ 2 \sqrt {y}-\ln \left (x \right )-c_{1} = 0 \] Verified OK.

\[ {}z^{\prime } = 10^{x +z} \]

\[ -\frac {10^{-z}}{\ln \left (10\right )}-\frac {10^{x}}{\ln \left (10\right )}-c_{1} = 0 \] Verified OK.

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

\[ \arctan \left (y\right )+\frac {\ln \left (1+y^{2}\right )}{2}+{\mathrm e}^{y}-\frac {{\mathrm e}^{2 x}}{2}-c_{1} = 0 \] Verified OK.

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

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

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

\[ \frac {y}{x}-\ln \left (\frac {y}{x}\right )-\ln \left (x \right )-c_{2} = 0 \] Verified OK.

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

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

\[ {}x y^{\prime } = y-x \,{\mathrm e}^{\frac {y}{x}} \]

\[ -{\mathrm e}^{-\frac {y}{x}}+\ln \left (x \right )-c_{2} = 0 \] Verified OK.

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

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

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

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

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

\[ \arcsin \left (\frac {y}{x}\right )-\ln \left (x \right )-c_{2} = 0 \] Verified OK. {0 < x}

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

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

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

\[ 0 = x +y \] Verified OK.

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

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

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

\[ \frac {y}{x}-\ln \left (\frac {y}{x}\right )-\ln \left (x \right )-c_{2} = 0 \] Verified OK.

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

\[ \arctan \left (\frac {y}{x}\right )-\ln \left (x \right )-c_{2} = 0 \] Verified OK.

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

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

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

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

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

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

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

\[ {\left (\frac {\sqrt {\frac {y^{2}-4 x y}{x^{2}}}\, x +y-2 x}{y}\right )}^{\frac {1}{4}} = c_{3} {\mathrm e}^{c_{2}} \sqrt {x} \] Verified OK. {0 < x}

\[ {\left (\frac {\sqrt {\frac {y^{2}-4 x y}{x^{2}}}\, x -y+2 x}{y}\right )}^{\frac {1}{4}} = c_{6} \sqrt {x}\, {\mathrm e}^{c_{5}} \] Verified OK. {0 < x}

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

\[ \operatorname {arcsinh}\left (\frac {y}{x}\right )-\ln \left (x \right )-c_{2} = 0 \] Verified OK. {0 < x}

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

\[ \sqrt {\frac {x^{2}+y^{2}}{x^{2}}}-1 = \frac {c_{3} {\mathrm e}^{c_{2}}}{x} \] Verified OK. {0 < x}

\[ \sqrt {\frac {x^{2}+y^{2}}{x^{2}}}+1 = \frac {c_{6} {\mathrm e}^{c_{5}}}{x} \] Verified OK. {0 < x}

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

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

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

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

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

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