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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ \frac {\sin \left (y\right ) \left (-\cos \left (y\right )+2 x \right )}{2}-\frac {y}{2} = \frac {1}{2}-\frac {\sqrt {3}}{8}-\frac {\pi }{12} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}r^{\prime \prime } = -\frac {k}{r^{2}} \]

\[ -\frac {\sqrt {2}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {r \left (c_{1} r+k \right )}}{r \sqrt {c_{1}}}\right ) k -\sqrt {r \left (c_{1} r+k \right )}\, \sqrt {c_{1}}\right )}{2 c_{1}^{\frac {3}{2}}} = t +c_{2} \] Verified OK.

\[ \frac {\sqrt {2}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {r \left (c_{1} r+k \right )}}{r \sqrt {c_{1}}}\right ) k -\sqrt {r \left (c_{1} r+k \right )}\, \sqrt {c_{1}}\right )}{2 c_{1}^{\frac {3}{2}}} = t +c_{3} \] Verified OK.

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

\[ \int _{}^{y}\frac {1}{\sqrt {\textit {\_a}^{3} k +2 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {1}{\sqrt {\textit {\_a}^{3} k +2 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime } = 2 k y^{3} \]

\[ \int _{}^{y}\frac {1}{\sqrt {\textit {\_a}^{4} k +2 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {1}{\sqrt {\textit {\_a}^{4} k +2 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}r^{\prime \prime } = \frac {h^{2}}{r^{3}}-\frac {k}{r^{2}} \]

\[ \frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{2 c_{1}}-\frac {k \ln \left (\frac {\left (2 c_{1} r+k \right ) \sqrt {2}}{2 \sqrt {c_{1}}}+\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}\right ) \sqrt {2}}{4 c_{1}^{\frac {3}{2}}} = t +c_{2} \] Verified OK.

\[ -\frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{2 c_{1}}+\frac {k \ln \left (\frac {\left (2 c_{1} r+k \right ) \sqrt {2}}{2 \sqrt {c_{1}}}+\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}\right ) \sqrt {2}}{4 c_{1}^{\frac {3}{2}}} = t +c_{3} \] Verified OK.

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

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

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

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

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

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

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

\[ \frac {2 \ln \left (3 \left (y-\frac {b}{3 a x}\right ) a x -b \right ) \sqrt {b^{2}+4 a}-\ln \left (9 a^{2} \left (y-\frac {b}{3 a x}\right )^{2} x^{2}+\left (3 b \left (y-\frac {b}{3 a x}\right ) x -9\right ) a -2 b^{2}\right ) \sqrt {b^{2}+4 a}+2 b \,\operatorname {arctanh}\left (\frac {6 \left (y-\frac {b}{3 a x}\right ) a x +b}{3 \sqrt {b^{2}+4 a}}\right )}{2 \sqrt {b^{2}+4 a}} = \ln \left (x \right )+c_{1} \] Verified OK.

\[ {}y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-a -1} = 0 \]

\[ \frac {-32 \left (a -1\right )^{\frac {-2+a}{a -1}} {\mathrm e}^{\frac {i \pi +2 x^{1-a}}{a -1}} \left (\left (x -\frac {x^{-1+2 a}}{4}\right ) 2^{\frac {-3 a +5}{a -1}}+\frac {x^{-1+2 a} 4^{\frac {1}{a -1}}}{32}\right ) \left (y+x^{-a}\right )^{2} \operatorname {WhittakerM}\left (-\frac {1}{a -1}, \frac {a -3}{2 a -2}, -\frac {4 x^{1-a}}{a -1}\right )-\left (\left (4 \left (y+x^{-a}\right )^{2} x^{2}+2 x^{a +1} \left (y+x^{-a}\right )^{2}+a +1\right ) {\mathrm e}^{\frac {4 x^{1-a}}{a -1}}+2 c_{1} \left (y+x^{-a}\right )^{2} \left (a +1\right )\right ) \left (a -3\right )}{2 \left (a +1\right ) \left (a -3\right ) \left (y+x^{-a}\right )^{2}} = 0 \] Warning, solution could not be verified

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

\[ y = 0 \] Verified OK.

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

\[ y = i x \] Verified OK.

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

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

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

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

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

\[ -\sqrt {\left (a +y+c_{2} \right ) \left (a -y-c_{2} \right )} = x +c_{5} \] Verified OK.

\[ \sqrt {-y^{2}-2 y c_{2} -c_{2}^{2}+a^{2}} = x +c_{6} \] Verified OK.

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

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

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

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

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

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

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

\[ -\sqrt {1-y^{2}}-\sqrt {-x^{2}+1}+\sqrt {3} = 0 \] Verified OK.

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

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

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

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

\[ {}y y^{\prime }+x y^{2}-8 x = 0 \]

\[ \sqrt {y^{2}-8} = {\mathrm e}^{-\frac {\left (x -1\right ) \left (x +1\right )}{2}} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ -\ln \left (x \right )-\frac {3 \ln \left (4 v^{2}-1\right )}{8} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ -\ln \left (y-1\right )+\ln \left (y-2\right ) = i \pi +x \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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