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

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

\[ {}y^{\prime } = \frac {2 y-x +5}{2 x -y-4} \]

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

\[ {}y^{\prime } = -\frac {4 x +3 y+15}{2 x +y+7} \]

\[ \left (4 x +13+y\right )^{2} \left (y+4+x \right ) = c_{4} {\mathrm e}^{3 c_{3}} \] Verified OK.

\[ {}y^{\prime } = \frac {x +3 y-5}{x -y-1} \]

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

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

\[ \frac {2 y}{5}-\frac {x}{5}-\frac {\ln \left (y+2 x -1\right )}{5}-c_{2} = 0 \] Verified OK.

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

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

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

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

\[ {}y^{\prime } = \frac {x -2 y+5}{y-2 x -4} \]

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

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

\[ \frac {\ln \left (\frac {\left (2+y\right )^{2}}{\left (1+x \right )^{2}}+\frac {6+3 y}{1+x}-3\right )}{2}-\frac {\sqrt {21}\, \operatorname {arctanh}\left (\frac {\left (2 y+7+3 x \right ) \sqrt {21}}{21+21 x}\right )}{21}+\ln \left (1+x \right )-c_{3} = 0 \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ y = -i a -i x \] Verified OK.

\[ y = i a +i x \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ \ln \left (y+1\right ) = x \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

\[ \sqrt {y} = x -x_{0} +\sqrt {y_{0}} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

\[ \ln \left (\frac {\left (y+\frac {3}{7}\right )^{2}}{\left (x -\frac {1}{7}\right )^{2}}+\frac {y+\frac {3}{7}}{x -\frac {1}{7}}+1\right )-\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (2 y+\frac {5}{7}+x \right ) \sqrt {3}}{3 x -\frac {3}{7}}\right )}{3}+2 \ln \left (x -\frac {1}{7}\right )-c_{3} = 0 \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ -\frac {1}{4 y^{4}}-\frac {1}{4 x^{4}}-c_{1} = 0 \] Verified OK.

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

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

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

\[ \frac {\left (-3 x^{3}+18 x \right ) \sin \left (x \right )}{3}+\frac {y^{3} x^{3}}{3}-3 x^{2} \cos \left (x \right )+6 \cos \left (x \right ) = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ \frac {2 y^{2} x +\frac {4}{5}}{x^{\frac {5}{2}}} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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