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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime } = \sqrt {25-y^{2}} \]

\[ \arcsin \left (\frac {y}{5}\right ) = t +4+\arcsin \left (\frac {3}{5}\right ) \] Verified OK.

\[ {}y^{\prime } = \sqrt {25-y^{2}} \]

\[ \arcsin \left (\frac {y}{5}\right ) = t -3-\arcsin \left (\frac {6}{5}\right ) \] Verified OK.

\[ {}y^{\prime }+\frac {y}{\sqrt {-t^{2}+4}} = t \]

\[ -\left (\int _{0}^{t}-\frac {\left (-\textit {\_a} \sqrt {-\textit {\_a}^{2}+4}+y\right ) {\mathrm e}^{\arcsin \left (\frac {\textit {\_a}}{2}\right )}}{\sqrt {-\textit {\_a}^{2}+4}}d \textit {\_a} \right )-y \left (\int _{0}^{t}\frac {{\mathrm e}^{\arcsin \left (\frac {\textit {\_a}}{2}\right )}}{\sqrt {-\textit {\_a}^{2}+4}}d \textit {\_a} \right )+{\mathrm e}^{\arcsin \left (\frac {t}{2}\right )} y = 0 \] Verified OK.

\[ {}y^{\prime }+\frac {y}{\sqrt {-t^{2}+4}} = t \]

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

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

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

\[ {}y^{\prime } = -\frac {t}{y} \]

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

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

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

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

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

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

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

\[ {}6+4 t^{3}+\left (5+\frac {9}{y^{8}}\right ) y^{\prime } = 0 \]

\[ -\frac {t^{4}}{2}-3 t -\frac {5 y}{2}+\frac {9}{14 y^{7}} = c_{1} \] Verified OK.

\[ {}\frac {6}{t^{9}}-\frac {6}{t^{3}}+t^{7}+\left (9+\frac {1}{s^{2}}-4 s^{8}\right ) s^{\prime } = 0 \]

\[ \frac {-t^{16}-24 t^{6}+6}{8 t^{8}}+\frac {4 s^{9}}{9}-9 s+\frac {1}{s} = c_{1} \] Verified OK.

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

\[ \frac {1}{t}+\frac {2 \sqrt {y}\, \left (y+3\right )}{9} = c_{1} \] Verified OK.

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

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

\[ {}\cos \left (y\right ) y^{\prime } = 8 \sin \left (8 t \right ) \]

\[ \frac {\cos \left (8 t \right )}{8}+\frac {\sin \left (y\right )}{8} = c_{1} \] Verified OK.

\[ {}\left (5 x^{5}-4 \cos \left (x\right )\right ) x^{\prime }+2 \cos \left (9 t \right )+2 \sin \left (7 t \right ) = 0 \]

\[ -\frac {\sin \left (9 t \right )}{9}+\frac {\cos \left (7 t \right )}{7}-\frac {5 x^{6}}{12}+2 \sin \left (x\right ) = c_{1} \] Verified OK.

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

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

\[ {}3 \sin \left (t \right )-\sin \left (3 t \right ) = \left (\cos \left (4 y\right )-4 \cos \left (y\right )\right ) y^{\prime } \]

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

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

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

\[ {}y^{\prime } = \frac {t^{3}}{y \sqrt {\left (1-y^{2}\right ) \left (t^{4}+9\right )}} \]

\[ -\frac {\left (t^{4}+9\right ) \sqrt {y+1}\, \sqrt {y-1}}{2 \sqrt {-\left (-1+y^{2}\right ) \left (t^{4}+9\right )}}+\frac {\left (y-1\right )^{\frac {3}{2}} \left (y+1\right )^{\frac {3}{2}}}{3} = c_{1} \] Verified OK.

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

\[ \frac {\cos \left (2 x \right )^{4}}{8}-\frac {\sec \left (y\right )^{2}}{2} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

\[ \frac {\left (2 \cos \left (x \right )-5\right )^{4}}{8}-\frac {\sec \left (y\right )^{3}}{3} = c_{1} \] Verified OK.

\[ {}\frac {\sqrt {\ln \left (x \right )}}{x} = \frac {{\mathrm e}^{\frac {3}{y}} y^{\prime }}{y} \]

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

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

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

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

\[ 2 \cos \left (t \right )+\ln \left (\sec \left (y\right )+\tan \left (y\right )\right ) = c_{1} \] Verified OK.

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

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

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

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

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

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

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

\[ \sin \left (y\right ) = x +\sin \left (2\right ) \] Verified OK.

\[ {}y^{\prime } = \frac {\sqrt {t}}{y} \]

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

\[ {}y^{\prime } = \sqrt {\frac {y}{t}} \]

\[ -\frac {2 \left (\sqrt {\frac {y}{t}}\, t -y\right )}{\sqrt {y}} = 2 \sqrt {2}-2 \] Verified OK.

\[ {}y^{\prime } = \frac {{\mathrm e}^{t}}{y+1} \]

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}\frac {t}{\sqrt {t^{2}+y^{2}}}+\frac {y y^{\prime }}{\sqrt {t^{2}+y^{2}}} = 0 \]

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

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

\[ -\frac {t^{2}}{2}-\frac {y^{2}}{6} = c_{1} \] Verified OK.

\[ {}t -\sin \left (t \right ) y+\left (y^{6}+\cos \left (t \right )\right ) y^{\prime } = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}3 t^{2}+3 y^{2}+6 t y y^{\prime } = 0 \]

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

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

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

\[ {}3 t^{2} y+3 y^{2}-1+\left (t^{3}+6 t y\right ) y^{\prime } = 0 \]

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

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

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

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

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

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

\[ \left (3+t \right ) \sin \left (t +y\right ) = c_{1} \] Verified OK.

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

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

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

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

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

\[ \frac {5 t^{2}}{2}-t y+t -y^{2} = 0 \] Verified OK.

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

\[ t y^{2}+\cos \left (2 t \right )+y = 2 \] Verified OK.

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

\[ y t +\frac {\sin \left (2 t \right )}{2}+\sec \left (y\right ) = 1 \] Verified OK.

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

\[ \arctan \left (t \right )-t y^{2} = 0 \] Verified OK.

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

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

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

\[ -x^{4}-\cos \left (6 x y\right )+y^{4} = -1 \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}\frac {9 t}{5}+2 y+\left (2 t +2 y\right ) y^{\prime } = 0 \]

\[ 2 t y+y^{2}+\frac {9 t^{2}}{10} = c_{1} \] Verified OK.

\[ {}2 t +\frac {19 y}{10}+\left (\frac {19 t}{10}+2 y\right ) y^{\prime } = 0 \]

\[ y^{2}+\frac {19 t y}{10}+t^{2} = c_{1} \] Verified OK.

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

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

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

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

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

\[ \frac {\left (9 \cos \left (t \right )-3 \sin \left (t \right )+20 y^{\frac {3}{2}}\right ) {\mathrm e}^{-3 t}}{30} = c_{1} \] Verified OK.

\[ {}y^{\prime }+3 y = \sqrt {y}\, \sin \left (t \right ) \]

\[ \sqrt {y} = \frac {3 \sin \left (t \right )}{13}-\frac {2 \cos \left (t \right )}{13}+c_{1} {\mathrm e}^{-\frac {3 t}{2}} \] Verified OK.

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

\[ \frac {1}{\sqrt {y}} = -\frac {t^{3}}{7}+\frac {c_{1}}{\sqrt {t}} \] Verified OK.

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

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

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

\[ \int _{}^{t}-\sin \left (2 \textit {\_a} \right ) \ln \left (\textit {\_a} \right )d \textit {\_a} -\frac {\sin \left (2 y\right ) \ln \left (y\right )}{2}-\frac {\pi }{4}+\frac {\operatorname {Si}\left (2 y\right )}{2} = c_{1} \] Verified OK. {y::positive}

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

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

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

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

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

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

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

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

\[ {}y^{\prime } = \frac {t +4 y}{4 t +y} \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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