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

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

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

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

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

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

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

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

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

\[ y = 0 \] Verified OK.

\[ x = \frac {\left (-8 x^{2}-2 y\right ) \sqrt {x^{2}+y}+8 x^{3}+6 y x +3 c_{1}}{3 \left (x -\sqrt {x^{2}+y}\right )^{2}} \] Verified OK.

\[ x = \frac {\left (8 x^{2}+2 y\right ) \sqrt {x^{2}+y}+8 x^{3}+6 y x +3 c_{1}}{3 \left (x +\sqrt {x^{2}+y}\right )^{2}} \] Verified OK.

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

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

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

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

\[ y = 0 \] 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.

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

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

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

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

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

\[ {}3 \,{\mathrm e}^{x} \tan \left (y\right )+\left (-{\mathrm e}^{x}+1\right ) \sec \left (y\right )^{2} y^{\prime } = 0 \]

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

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

\[ \int _{}^{x}\frac {2}{\sqrt {2 \textit {\_a}^{4}-4 \textit {\_a}^{2}+8 c_{1} +2}}d \textit {\_a} = t +c_{2} \] Verified OK.

\[ \int _{}^{x}-\frac {2}{\sqrt {2 \textit {\_a}^{4}-4 \textit {\_a}^{2}+8 c_{1} +2}}d \textit {\_a} = t +c_{3} \] Verified OK.

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

\[ \int _{}^{x}\frac {2}{\sqrt {-2 \textit {\_a}^{4}-4 \textit {\_a}^{2}+8 c_{1} -2}}d \textit {\_a} = t +c_{2} \] Verified OK.

\[ \int _{}^{x}-\frac {2}{\sqrt {-2 \textit {\_a}^{4}-4 \textit {\_a}^{2}+8 c_{1} -2}}d \textit {\_a} = t +c_{3} \] Verified OK.

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

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

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

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

\[ \frac {\left (\left \{\begin {array}{cc} -2 \sqrt {-y} & y\le 0 \\ 2 \sqrt {y} & 0

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

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

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

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

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

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

\[ {}y^{\prime } = \frac {3 y}{\left (x -5\right ) \left (x +3\right )}+{\mathrm e}^{-x} \]

\[ \int _{}^{x}-\frac {\left (\textit {\_a}^{2}-2 \textit {\_a} -15\right ) {\mathrm e}^{-\textit {\_a}}+3 y}{\left (\textit {\_a} -5\right )^{\frac {11}{8}} \left (3+\textit {\_a} \right )^{\frac {5}{8}}}d \textit {\_a} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime } = \sqrt {\frac {y-4}{x}} \]

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

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

\[ \frac {4 \left (y x \right )^{\frac {3}{4}} \left (7 y^{\frac {3}{4}}-3 x \right )}{21 y^{\frac {3}{4}}} = c_{1} \] Verified OK.

\[ {}y^{\prime } = 4 y-5 \]

\[ -\frac {\ln \left (2\right )}{2}+\frac {\ln \left (4 y-5\right )}{4} = x -1+\frac {\ln \left (11\right )}{4}-\frac {\ln \left (2\right )}{2} \] Verified OK.

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

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

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

\[ \frac {\ln \left (a y+b \right )}{a} = \frac {\ln \left (a d +b \right )+\left (-c +x \right ) a}{a} \] Verified OK.

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

\[ \frac {\ln \left (y\right )}{3} = x +\frac {i \pi }{3} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime } = 4 y+1 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ -\frac {\ln \left (t -1\right )}{2}-\frac {\ln \left (1+t \right )}{2}-\frac {y^{2}}{2} = -\frac {i \pi }{2}-8 \] Verified OK.

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

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

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

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

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

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

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

\[ y \left (y+3\right ) = 4+t \] Verified OK.

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

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

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

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

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

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

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

\[ -\frac {1}{S-1}-\ln \left (S-1\right )+\ln \left (S\right ) = -i \pi +t +1 \] Verified OK.

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

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

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

\[ -\frac {1}{S-1}-\ln \left (S-1\right )+\ln \left (S\right ) = t +\frac {2}{3}-\ln \left (3\right ) \] 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.

\[ {}v^{\prime } = 2 V \left (t \right )-2 v \]

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

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

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

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

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

\[ {}w^{\prime } = \left (3-w\right ) \left (w+1\right ) \]

\[ \frac {\ln \left (w+1\right )}{4}-\frac {\ln \left (w-3\right )}{4} = t +\frac {\ln \left (5\right )}{4} \] Verified OK.

\[ {}w^{\prime } = \left (3-w\right ) \left (w+1\right ) \]

\[ \frac {\ln \left (w+1\right )}{4}-\frac {\ln \left (w-3\right )}{4} = t -\frac {\ln \left (3\right )}{4}-\frac {i \pi }{4} \] Verified OK.

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

\[ y \,{\mathrm e}^{-\frac {2}{y}}-2 \,\operatorname {expIntegral}_{1}\left (\frac {2}{y}\right ) = t +2 \,{\mathrm e}^{-1}-2 \,\operatorname {expIntegral}_{1}\left (1\right ) \] Verified OK.

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

\[ y \,{\mathrm e}^{-\frac {2}{y}}-2 \,\operatorname {expIntegral}_{1}\left (\frac {2}{y}\right ) = t -1+2 \,{\mathrm e}^{-1}-2 \,\operatorname {expIntegral}_{1}\left (1\right ) \] Verified OK.

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

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

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

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

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

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

\[ {}\theta ^{\prime } = \frac {9}{10}-\frac {11 \cos \left (\theta \right )}{10} \]

\[ -\sqrt {10}\, \operatorname {arctanh}\left (\tan \left (\frac {\theta }{2}\right ) \sqrt {10}\right ) = t -\sqrt {10}\, \operatorname {arccoth}\left (\tan \left (\frac {1}{2}\right ) \sqrt {10}\right )+\frac {i \pi \sqrt {10}}{2} \] Verified OK.

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

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

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

\[ \frac {\ln \left (y\right )}{3}-\frac {\ln \left (y-1\right )}{2}+\frac {\ln \left (y-3\right )}{6} = t +\frac {\ln \left (2\right )}{3}+\frac {i \pi }{6} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

\[ \frac {\ln \left (y-2\right )}{6}-\frac {\ln \left (y\right )}{6} = t +\frac {i \pi }{6} \] Verified OK.

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

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

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

\[ \frac {\ln \left (y-2\right )}{6}-\frac {\ln \left (y\right )}{6} = t -\frac {\ln \left (3\right )}{6} \] Verified OK.

\[ {}y^{\prime } = y^{2}-4 y-12 \]

\[ \frac {\ln \left (y-6\right )}{8}-\frac {\ln \left (y+2\right )}{8} = t +\frac {\ln \left (5\right )}{8}+\frac {i \pi }{8}-\frac {\ln \left (3\right )}{8} \] Verified OK.

\[ {}y^{\prime } = y^{2}-4 y-12 \]

\[ \frac {\ln \left (y-6\right )}{8}-\frac {\ln \left (y+2\right )}{8} = t -1+\frac {i \pi }{8}+\frac {\ln \left (3\right )}{8} \] Verified OK.

\[ {}y^{\prime } = y^{2}-4 y-12 \]

\[ \frac {\ln \left (y-6\right )}{8}-\frac {\ln \left (y+2\right )}{8} = t +\frac {i \pi }{8}-\frac {\ln \left (7\right )}{8} \] Verified OK.

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

\[ \ln \left (\sec \left (y\right )+\tan \left (y\right )\right ) = t \] Verified OK.

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

\[ \ln \left (\sec \left (y\right )+\tan \left (y\right )\right ) = t +1+\ln \left (\sin \left (1\right )+1\right )-\ln \left (\cos \left (1\right )\right ) \] Verified OK.

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

\[ \ln \left (\sec \left (y\right )+\tan \left (y\right )\right ) = i \pi +t \] Verified OK.