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

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

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

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

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

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

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

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

\[ {}y^{\prime } = 3 \sqrt {x y} \]

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

\[ {}y^{\prime } = 4 \left (x y\right )^{\frac {1}{3}} \]

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}2 y y^{\prime } = \frac {x}{\sqrt {x^{2}-16}} \]

\[ y^{2}-\sqrt {x^{2}-16} = 1 \] Verified OK.

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

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

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

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

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

\[ -\ln \left (y-2\right ) = x -\ln \left (2\right )-i \pi \] Verified OK.

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

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

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

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

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

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

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

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

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

\[ \frac {2 y^{2}+x^{2}}{4 x^{6}} = c_{1} \] Verified OK.

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

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

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

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

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

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

\[ {}2 x y^{3}+y^{2} y^{\prime } = 6 x \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}\frac {2 x}{y}-\frac {3 y^{2}}{x^{4}}+\left (-\frac {x^{2}}{y^{2}}+\frac {1}{\sqrt {y}}+\frac {2 y}{x^{3}}\right ) y^{\prime } = 0 \]

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

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

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

\[ {}2 y+x y^{\prime } = 6 x^{2} \sqrt {y} \]

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

\[ {}6 x y^{3}+2 y^{4}+\left (9 x^{2} y^{2}+8 x y^{3}\right ) y^{\prime } = 0 \]

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

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

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

\[ {}x y^{\prime } = 12 x^{4} y^{\frac {2}{3}}+6 y \]

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

\[ {}9 \sqrt {x}\, y^{\frac {4}{3}}-12 x^{\frac {1}{5}} y^{\frac {3}{2}}+\left (8 x^{\frac {3}{2}} y^{\frac {1}{3}}-15 x^{\frac {6}{5}} \sqrt {y}\right ) y^{\prime } = 0 \]

\[ 6 y^{\frac {4}{3}} x^{\frac {3}{2}}-10 y^{\frac {3}{2}} x^{\frac {6}{5}} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

\[ \arctan \left (y\right ) = x \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ -\frac {\left (x^{2}+1\right )^{2}}{4}+y^{4} = 0 \] Verified OK.

\[ {}y^{\prime } = \frac {-{\mathrm e}^{x}+3 x^{2}}{-5+2 y} \]

\[ x^{3}-y^{2}-{\mathrm e}^{x}+5 y = 3 \] Verified OK.

\[ {}y^{\prime } = \frac {{\mathrm e}^{-x}-{\mathrm e}^{x}}{3+4 y} \]

\[ -2 y^{2}-{\mathrm e}^{x}-3 y-{\mathrm e}^{-x} = -7 \] Verified OK.

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

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

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

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

\[ {}y^{\prime } = \frac {3 x^{2}+1}{-6 y+3 y^{2}} \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y+\ln \left (t \right ) y^{\prime } = \cot \left (t \right ) \]

\[ \int _{}^{t}\frac {\left (y-\cot \left (\textit {\_a} \right )\right ) {\mathrm e}^{-\operatorname {expIntegral}_{1}\left (-\ln \left (\textit {\_a} \right )\right )}}{\ln \left (\textit {\_a} \right )}d \textit {\_a} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {t^{2}+1}{3 y-y^{2}} \]

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

\[ {}y^{\prime } = -\frac {4 t}{y} \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ \frac {a \,x^{2}}{2}+b y x +\frac {c y^{2}}{2} = c_{1} \] Verified OK.

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

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

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

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

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

\[ y^{2}-y x +x^{2} = 7 \] Verified OK.

\[ {}-1+9 x^{2}+y+\left (x -4 y\right ) y^{\prime } = 0 \]

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

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

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

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

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

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

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

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

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

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

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