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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime } = \sqrt {\frac {9 y^{2}-6 y+2}{x^{2}-2 x +5}} \]

\[ -\frac {\sqrt {\frac {9 y^{2}-6 y+2}{x^{2}-2 x +5}}\, \sqrt {x^{2}-2 x +5}\, \operatorname {arcsinh}\left (-\frac {1}{2}+\frac {x}{2}\right )}{\sqrt {9 y^{2}-6 y+2}}+\frac {\operatorname {arcsinh}\left (3 y-1\right )}{3} = c_{1} \] Verified OK.

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

\[ y = 0 \] Verified OK.

\[ y = x +1 \] Verified OK.

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

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

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

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

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

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

\[ \frac {y^{2}}{2}-\frac {x^{2}}{4}-\frac {c_{1} x}{2}-c_{2} = 0 \] Warning, solution could not be verified

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

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

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

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

\[ \int _{}^{y}\frac {4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}}{\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}}+4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}+\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}}-16 i \sqrt {3}-8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ \int _{}^{y}\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}-16-\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {4}{3}}+8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 c_{1}^{2}+18 c_{1} \textit {\_a} +9 \textit {\_a}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = x +c_{4} \] Verified OK.

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

\[ -{\mathrm e}^{-y}-{\mathrm e}^{x} c_{2} -c_{3} = 0 \] Verified OK.

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

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

\[ \sqrt {-y^{2}-2 y c_{1} -c_{1}^{2}+9} = x +c_{3} \] Verified OK.

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

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

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

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

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

\[ \int _{}^{x}-\frac {2 \,{\mathrm e}^{2 \textit {\_a}}}{\sqrt {-2 \,{\mathrm e}^{2 \textit {\_a}} \left (2 \,{\mathrm e}^{2 \textit {\_a}} \textit {\_a} -{\mathrm e}^{2 \textit {\_a}}-4 c_{1} \right )}}d \textit {\_a} = t +c_{3} \] Verified OK.

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

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

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

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

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

\[ \frac {y^{2}}{2}+\frac {x^{2}}{4}-\frac {7 x}{4}-\frac {1}{2} = 0 \] Warning, solution could not be verified