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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ -\frac {x^{2} {\mathrm e}^{-2 y}}{2}-\frac {{\mathrm e}^{-2 y} \left (4 y^{3}+6 y^{2}+10 y+5\right )}{8} = c_{1} \] 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.

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

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

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

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

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

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

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

\[ \left \{\begin {array}{cc} -\ln \left (1-y\right ) & y\le 0 \\ \ln \left (1+y\right ) & 0

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ -\frac {x^{3}}{3}-\frac {3 x^{2}}{2}+\frac {y^{2}}{2}-2 x -2 y = -{\frac {23}{6}} \] Verified OK.

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

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

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

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

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

\[ -\frac {x^{2}}{2}+\frac {1}{6 y-6}+\frac {\ln \left (y-1\right )}{18}-\frac {\ln \left (2+y\right )}{18} = \frac {1}{12}+\frac {\ln \left (2\right )}{18}-\frac {\ln \left (5\right )}{18} \] Verified OK.

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

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

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

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

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

\[ -\frac {x^{2}}{2}-\frac {y^{2}}{2} = -{\frac {25}{2}} \] Verified OK.

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

\[ -\frac {\ln \left (3+y\right )}{9}+\frac {1}{3 y}+\frac {\ln \left (y\right )}{9}-\frac {x^{3}}{3}-c_{2} = 0 \] Verified OK.

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

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

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

\[ -\sin \left (x \right )-\cos \left (y\right ) = 0 \] Verified OK.

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

\[ \frac {\ln \left (y\right )-\ln \left (b y-a \right )}{a} = x +\frac {\ln \left (\operatorname {y0} \right )-\ln \left (b \operatorname {y0} -a \right )}{a} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}x^{2} y^{\prime }+2 y = 2 \,{\mathrm e}^{\frac {1}{x}} \sqrt {y} \]

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

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

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

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

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

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

\[ \frac {1}{y^{3}} = -3 x^{4}+11 x^{3} \] Verified OK.

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

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

\[ {}y^{\prime }-4 y = \frac {48 x}{y^{2}} \]

\[ y^{3} = -1+2 \,{\mathrm e}^{12 x}-12 x \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

\[ \frac {y^{3}}{3 x^{3}}-\ln \left (x \right ) = 9 \] Verified OK.

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

\[ \frac {x^{4}}{4}+\frac {x^{2} y^{2}}{2} = {\frac {9}{4}} \] Verified OK.

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

\[ \frac {x^{2}+y^{2}}{2 x^{8}} = 2 \] 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.

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

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

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

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

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

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

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

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

\[ {}y^{\prime } = \frac {-x +3 y-14}{x +y-2} \]

\[ \ln \left (\frac {y-6-x}{2+x}\right )+\frac {4+2 x}{-y+6+x}+\ln \left (2+x \right )-c_{3} = 0 \] Verified OK.

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

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

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

\[ \frac {x^{6}+y^{2}}{2 x^{12}} = c_{1} \] Verified OK.

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

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

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

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

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

\[ \frac {x y^{2}}{2}+\frac {1}{x}+\frac {3 y}{2} = {\frac {15}{2}} \] Verified OK.

\[ {}y^{\prime }+\frac {3 y}{x} = \frac {3 x^{4} y^{2}+10 x^{2} y+6}{x^{3} \left (2 x^{2} y+5\right )} \]

\[ y^{2} x^{3}+5 y x +\frac {6}{x} = 12 \] Verified OK.

\[ {}3 \cos \left (x \right ) y+4 x \,{\mathrm e}^{x}+2 x^{3} y+\left (3 \sin \left (x \right )+3\right ) y^{\prime } = 0 \]

\[ \int _{}^{x}\left (2 y \textit {\_a}^{3}+4 \textit {\_a} \,{\mathrm e}^{\textit {\_a}}+3 \cos \left (\textit {\_a} \right ) y\right ) {\mathrm e}^{\frac {12 \textit {\_a}^{2} \left ({\mathrm e}^{i \textit {\_a}}+i\right ) \ln \left (1-i {\mathrm e}^{i \textit {\_a}}\right )+\left (-24 i {\mathrm e}^{i \textit {\_a}} \textit {\_a} +24 \textit {\_a} \right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i \textit {\_a}}\right )+\left (24 i+24 \,{\mathrm e}^{i \textit {\_a}}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i \textit {\_a}}\right )-4 i {\mathrm e}^{i \textit {\_a}} \textit {\_a}^{3}}{3 \,{\mathrm e}^{i \textit {\_a}}+3 i}}d \textit {\_a} +\left (-\left (\int _{}^{x}\left (2 \textit {\_a}^{3}+3 \cos \left (\textit {\_a} \right )\right ) {\mathrm e}^{\frac {12 \textit {\_a}^{2} \left ({\mathrm e}^{i \textit {\_a}}+i\right ) \ln \left (1-i {\mathrm e}^{i \textit {\_a}}\right )+\left (-24 i {\mathrm e}^{i \textit {\_a}} \textit {\_a} +24 \textit {\_a} \right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i \textit {\_a}}\right )+\left (24 i+24 \,{\mathrm e}^{i \textit {\_a}}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i \textit {\_a}}\right )-4 i {\mathrm e}^{i \textit {\_a}} \textit {\_a}^{3}}{3 \,{\mathrm e}^{i \textit {\_a}}+3 i}}d \textit {\_a} \right )+\left (3 \sin \left (x \right )+3\right ) {\mathrm e}^{\frac {12 x^{2} \left ({\mathrm e}^{i x}+i\right ) \ln \left (1-i {\mathrm e}^{i x}\right )+\left (-24 i {\mathrm e}^{i x} x +24 x \right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+\left (24 i+24 \,{\mathrm e}^{i x}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )-4 i {\mathrm e}^{i x} x^{3}}{3 \,{\mathrm e}^{i x}+3 i}}\right ) y = c_{1} \] Verified OK.

\[ {}2 x -2 y^{2}+\left (12 y^{2}-4 x y\right ) y^{\prime } = 0 \]

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

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

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

\[ {}3 x^{2}+2 x y+4 y^{2}+\left (x^{2}+8 x y+18 y\right ) y^{\prime } = 0 \]

\[ x^{3}+x^{2} y+4 x y^{2}+9 y^{2} = c_{1} \] 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.

\[ {}{\mathrm e}^{x} \left (x^{2} y^{2}+2 x y^{2}\right )+6 x +\left (2 x^{2} y \,{\mathrm e}^{x}+2\right ) y^{\prime } = 0 \]

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

\[ {}x^{2} {\mathrm e}^{y+x^{2}} \left (2 x^{2}+3\right )+4 x +\left (x^{3} {\mathrm e}^{y+x^{2}}-12 y^{2}\right ) y^{\prime } = 0 \]

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

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

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

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

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

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

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

\[ {}7 x +4 y+\left (4 x +3 y\right ) y^{\prime } = 0 \]

\[ \frac {x \left (7 x +8 y\right )}{2}+\frac {3 y^{2}}{2} = c_{1} \] Verified OK.

\[ {}{\mathrm e}^{x} \left (x^{4} y^{2}+4 x^{3} y^{2}+1\right )+\left (2 x^{4} y \,{\mathrm e}^{x}+2 y\right ) y^{\prime } = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

\[ -4 x^{2}+\frac {3 y}{x^{3}}+\frac {y^{2}}{x^{6}}+\frac {3 y}{x} = 3 \] Verified OK.

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

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

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

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

\[ {}27 x y^{2}+8 y^{3}+\left (18 x^{2} y+12 x y^{2}\right ) y^{\prime } = 0 \]

\[ y^{2} x^{2} \left (4 y+9 x \right ) = c_{1} \] Verified OK.

\[ {}6 x y^{2}+2 y+\left (12 x^{2} y+6 x +3\right ) y^{\prime } = 0 \]

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

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

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

\[ {}12 x^{3} y+24 x^{2} y^{2}+\left (9 x^{4}+32 x^{3} y+4 y\right ) y^{\prime } = 0 \]

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

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

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

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

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

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

\[ \int _{}^{x}\left (-\cos \left (\textit {\_a} \right ) a +\sin \left (\textit {\_a} \right ) y\right ) {\mathrm e}^{-i \textit {\_a} +2 \left (\int \frac {i \textit {\_a} -b}{\left (i b -\textit {\_a} \right ) {\mathrm e}^{2 i \textit {\_a}}+i b +\textit {\_a}}d \textit {\_a} \right )}d \textit {\_a} +\int _{0}^{y}\left (\sin \left (x \right ) {\mathrm e}^{-i x +2 \left (\int \frac {i x -b}{\left (i b -x \right ) {\mathrm e}^{2 i x}+i b +x}d x \right )} x -\cos \left (x \right ) {\mathrm e}^{-i x +2 \left (\int \frac {i x -b}{\left (i b -x \right ) {\mathrm e}^{2 i x}+i b +x}d x \right )} b -\left (\int _{}^{x}\sin \left (\textit {\_a} \right ) {\mathrm e}^{-i \textit {\_a} +2 \left (\int \frac {i b +\textit {\_a}}{{\mathrm e}^{2 i \textit {\_a}} \left (i \textit {\_a} +b \right )+b -i \textit {\_a}}d \textit {\_a} \right )}d \textit {\_a} \right )\right )d \textit {\_a} = c_{1} \] Warning, solution could not be verified

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

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

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

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