\[ {}\left \{x^{\prime }\relax (t ) = -y \relax (t )+\left (\left \{\begin {array}{cc} x \relax (t ) \left (x \relax (t )^{2}+y \relax (t )^{2}-1\right ) \sin \left (\frac {1}{x \relax (t )^{2}+y \relax (t )^{2}}\right ) & x \relax (t )^{2}+y \relax (t )^{2}\neq 1 \\ 0 & \mathit {otherwise} \end {array}\right .\right ), y^{\prime }\relax (t ) = x \relax (t )+\left (\left \{\begin {array}{cc} y \relax (t ) \left (x \relax (t )^{2}+y \relax (t )^{2}-1\right ) \sin \left (\frac {1}{x \relax (t )^{2}+y \relax (t )^{2}}\right ) & x \relax (t )^{2}+y \relax (t )^{2}\neq 1 \\ 0 & \mathit {otherwise} \end {array}\right .\right )\right \} \]

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

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

\[ {}\{x^{\prime }\relax (t ) y^{\prime }\relax (t )+t y^{\prime }\relax (t )-y \relax (t ) = 0, x^{\prime }\relax (t )^{2}+t x^{\prime }\relax (t )+a y^{\prime }\relax (t )-x \relax (t ) = 0\} \]

\[ {}\{x \relax (t ) = t x^{\prime }\relax (t )+f \left (x^{\prime }\relax (t ), y^{\prime }\relax (t )\right ), y \relax (t ) = t y^{\prime }\relax (t )+g \left (x^{\prime }\relax (t ), y^{\prime }\relax (t )\right )\} \]

\[ {}\left \{x^{\prime \prime }\relax (t ) = a \,{\mathrm e}^{2 x \relax (t )}-{\mathrm e}^{-x \relax (t )}+{\mathrm e}^{-2 x \relax (t )} \left (\cos ^{2}\left (y \relax (t )\right )\right ), y^{\prime \prime }\relax (t ) = {\mathrm e}^{-2 x \relax (t )} \sin \left (y \relax (t )\right ) \cos \left (y \relax (t )\right )-\frac {\sin \left (y \relax (t )\right )}{\cos \left (y \relax (t )\right )^{3}}\right \} \]

\[ {}\left \{x^{\prime \prime }\relax (t ) = \frac {k x \relax (t )}{\left (x \relax (t )^{2}+y \relax (t )^{2}\right )^{\frac {3}{2}}}, y^{\prime \prime }\relax (t ) = \frac {k y \relax (t )}{\left (x \relax (t )^{2}+y \relax (t )^{2}\right )^{\frac {3}{2}}}\right \} \]

\[ {}\left \{x^{\prime \prime }\relax (t ) = -\frac {C \left (y \relax (t )\right ) f \left (\sqrt {y^{\prime }\relax (t )^{2}}\right ) x^{\prime }\relax (t )}{\sqrt {x^{\prime }\relax (t )^{2}+y^{\prime }\relax (t )^{2}}}, y^{\prime \prime }\relax (t ) = -\frac {C \left (y \relax (t )\right ) f \left (\sqrt {y^{\prime }\relax (t )^{2}}\right ) y^{\prime }\relax (t )}{\sqrt {x^{\prime }\relax (t )^{2}+y^{\prime }\relax (t )^{2}}}-g\right \} \]

\[ {}\{x^{\prime }\relax (t ) = y \relax (t )-z \relax (t ), y^{\prime }\relax (t ) = x \relax (t )^{2}+y \relax (t ), z^{\prime }\relax (t ) = x \relax (t )^{2}+z \relax (t )\} \]

\[ {}\{a x^{\prime }\relax (t ) = \left (-c +b \right ) y \relax (t ) z \relax (t ), b y^{\prime }\relax (t ) = \left (c -a \right ) z \relax (t ) x \relax (t ), c z^{\prime }\relax (t ) = \left (a -b \right ) x \relax (t ) y \relax (t )\} \]

\[ {}\{x^{\prime }\relax (t ) = x \relax (t ) \left (y \relax (t )-z \relax (t )\right ), y^{\prime }\relax (t ) = y \relax (t ) \left (z \relax (t )-x \relax (t )\right ), z^{\prime }\relax (t ) = z \relax (t ) \left (x \relax (t )-y \relax (t )\right )\} \]

\[ {}\{x^{\prime }\relax (t )+y^{\prime }\relax (t ) = x \relax (t ) y \relax (t ), x^{\prime }\relax (t )+z^{\prime }\relax (t ) = x \relax (t ) z \relax (t ), y^{\prime }\relax (t )+z^{\prime }\relax (t ) = y \relax (t ) z \relax (t )\} \]

\[ {}\left \{x^{\prime }\relax (t ) = \frac {x \relax (t )^{2}}{2}-\frac {y \relax (t )}{24}, y^{\prime }\relax (t ) = 2 x \relax (t ) y \relax (t )-3 z \relax (t ), z^{\prime }\relax (t ) = 3 x \relax (t ) z \relax (t )-\frac {y \relax (t )^{2}}{6}\right \} \]

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

\[ {}\{x^{\prime }\relax (t ) = x \relax (t ) \left (y \relax (t )^{2}-z \relax (t )^{2}\right ), y^{\prime }\relax (t ) = -y \relax (t ) \left (z \relax (t )^{2}+x \relax (t )^{2}\right ), z^{\prime }\relax (t ) = z \relax (t ) \left (x \relax (t )^{2}+y \relax (t )^{2}\right )\} \]

\[ {}\{x^{\prime }\relax (t ) = -x \relax (t ) y \relax (t )^{2}+x \relax (t )+y \relax (t ), y^{\prime }\relax (t ) = x \relax (t )^{2} y \relax (t )-x \relax (t )-y \relax (t ), z^{\prime }\relax (t ) = y \relax (t )^{2}-x \relax (t )^{2}\} \]

\[ {}\{\left (x \relax (t )-y \relax (t )\right ) \left (x \relax (t )-z \relax (t )\right ) x^{\prime }\relax (t ) = f \relax (t ), \left (y \relax (t )-x \relax (t )\right ) \left (y \relax (t )-z \relax (t )\right ) y^{\prime }\relax (t ) = f \relax (t ), \left (z \relax (t )-x \relax (t )\right ) \left (z \relax (t )-y \relax (t )\right ) z^{\prime }\relax (t ) = f \relax (t )\} \]

\[ {}\{x_{1}^{\prime }\relax (t ) \sin \left (x_{2} \relax (t )\right ) = x_{4} \relax (t ) \sin \left (x_{3} \relax (t )\right )+x_{5} \relax (t ) \cos \left (x_{3} \relax (t )\right ), x_{3}^{\prime }\relax (t )+x_{1}^{\prime }\relax (t ) \cos \left (x_{2} \relax (t )\right ) = a, x_{4}^{\prime }\relax (t )-\left (1-\lambda \right ) a x_{5} \relax (t ) = -m \sin \left (x_{2} \relax (t )\right ) \cos \left (x_{3} \relax (t )\right ), x_{5}^{\prime }\relax (t )+\left (1-\lambda \right ) a x_{4} \relax (t ) = m \sin \left (x_{2} \relax (t )\right ) \sin \left (x_{3} \relax (t )\right ), x_{2}^{\prime }\relax (t ) = x_{4} \relax (t ) \cos \left (x_{3} \relax (t )\right )-x_{5} \relax (t ) \sin \left (x_{3} \relax (t )\right )\} \]

\[ {}y^{\prime } = f \relax (x ) \]

\[ {}y^{\prime } = f \relax (y) \]

\[ {}y^{\prime } = f \relax (x ) g \relax (y) \]

\[ {}g \relax (x ) y^{\prime } = f_{1}\relax (x ) y+f_{0} \relax (x ) \]

\[ {}g \relax (x ) y^{\prime } = f_{1}\relax (x ) y+f_{n} \relax (x ) y^{n} \]

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

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

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

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

\[ {}y^{\prime } = a y^{2}+b \,x^{n} \]

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

\[ {}y^{\prime } = a y^{2}+b \,x^{2 n}+c \,x^{n -1} \]

\[ {}y^{\prime } = a \,x^{n} y^{2}+b \,x^{-n -2} \]

\[ {}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m} \]

\[ {}y^{\prime } = y^{2}+k \left (a x +b \right )^{n} \left (c x +d \right )^{-n -4} \]

\[ {}y^{\prime } = a \,x^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{n +2 m} \]

\[ {}y^{\prime } = \left (a \,x^{2 n}+b \,x^{n -1}\right ) y^{2}+c \]

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

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