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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}{y^{\prime }}^{2}-5 y^{\prime }+6 = 0 \]

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

\[ {}{y^{\prime }}^{2}-7 y^{\prime }+12 = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ y = c_{3} x \] Verified OK.

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

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

\[ {}{y^{\prime }}^{2} x^{2}+4 x y y^{\prime }-5 y^{2} = 0 \]

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

\[ {}{y^{\prime }}^{2} x^{2}-5 x y y^{\prime }+6 y^{2} = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}{y^{\prime }}^{3}-7 y^{\prime }+6 = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ y = c_{3} x \] Verified OK.

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

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

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

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

\[ {}{y^{\prime }}^{2}-5 y^{\prime }+6 = 0 \]

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

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

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

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

\[ y = -\tanh \left (-\operatorname {arctanh}\left (x \right )+c_{2} \right ) \] Verified OK.

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

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

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

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

\[ {}x {y^{\prime }}^{5}-y {y^{\prime }}^{4}+\left (x^{2}+1\right ) {y^{\prime }}^{3}-2 x y {y^{\prime }}^{2}+\left (x +y^{2}\right ) y^{\prime }-y = 0 \]

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

\[ y = -\frac {2 \sqrt {3}\, \left (-x \right )^{\frac {3}{2}}}{9} \] Verified OK.

\[ y = \frac {2 \sqrt {3}\, \left (-x \right )^{\frac {3}{2}}}{9} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

\[ {}{y^{\prime }}^{2} x^{2}-5 x y y^{\prime }+6 y^{2} = 0 \]

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

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

\[ y = c_{3} x \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ y = c_{3} x \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}a y y^{\prime \prime }+b y = 0 \]

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

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

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