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

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

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

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

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

\[ {}[x^{\prime }\left (t \right )-y^{\prime }\left (t \right )+x \left (t \right ) = 2 t, x^{\prime \prime }\left (t \right )+y^{\prime }\left (t \right )-9 x \left (t \right )+3 y \left (t \right ) = \sin \left (2 t \right )] \]

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

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

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

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

\[ {}[x^{\prime \prime }\left (t \right ) = a_{1} x \left (t \right )+b_{1} y \left (t \right )+c_{1}, y^{\prime \prime }\left (t \right ) = a_{2} x \left (t \right )+b_{2} y \left (t \right )+c_{2}] \]

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

\[ {}\left [x^{\prime \prime }\left (t \right ) = \left (3 \cos \left (a t +b \right )^{2}-1\right ) c^{2} x \left (t \right )+\frac {3 c^{2} y \left (t \right ) \sin \left (2 a t b \right )}{2}, y^{\prime \prime }\left (t \right ) = \left (3 \sin \left (a t +b \right )^{2}-1\right ) c^{2} y \left (t \right )+\frac {3 c^{2} x \left (t \right ) \sin \left (2 a t b \right )}{2}\right ] \]

\[ {}[x^{\prime \prime }\left (t \right )+6 x \left (t \right )+7 y \left (t \right ) = 0, y^{\prime \prime }\left (t \right )+3 x \left (t \right )+2 y \left (t \right ) = 2 t] \]

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

\[ {}[a_{1} x^{\prime \prime }\left (t \right )+b_{1} x^{\prime }\left (t \right )+c_{1} x \left (t \right )-A y^{\prime }\left (t \right ) = B \,{\mathrm e}^{i \omega t}, a_{2} y^{\prime \prime }\left (t \right )+b_{2} y^{\prime }\left (t \right )+c_{2} y \left (t \right )+A x^{\prime }\left (t \right ) = 0] \]

\[ {}[x^{\prime \prime }\left (t \right )+a \left (x^{\prime }\left (t \right )-y^{\prime }\left (t \right )\right )+b_{1} x \left (t \right ) = c_{1} {\mathrm e}^{i \omega t}, y^{\prime \prime }\left (t \right )+a \left (y^{\prime }\left (t \right )-x^{\prime }\left (t \right )\right )+b_{2} y \left (t \right ) = c_{2} {\mathrm e}^{i \omega t}] \]

\[ {}[\operatorname {a11} x^{\prime \prime }\left (t \right )+\operatorname {b11} x^{\prime }\left (t \right )+\operatorname {c11} x \left (t \right )+\operatorname {a12} y^{\prime \prime }\left (t \right )+\operatorname {b12} y^{\prime }\left (t \right )+\operatorname {c12} y \left (t \right ) = 0, \operatorname {a21} x^{\prime \prime }\left (t \right )+\operatorname {b21} x^{\prime }\left (t \right )+\operatorname {c21} x \left (t \right )+\operatorname {a22} y^{\prime \prime }\left (t \right )+\operatorname {b22} y^{\prime }\left (t \right )+\operatorname {c22} y \left (t \right ) = 0] \]

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

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

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

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

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

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

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

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

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

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

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

\[ {}[x^{\prime }\left (t \right ) = -3 x \left (t \right )+48 y \left (t \right )-28 z \left (t \right ), y^{\prime }\left (t \right ) = -4 x \left (t \right )+40 y \left (t \right )-22 z \left (t \right ), z^{\prime }\left (t \right ) = -6 x \left (t \right )+57 y \left (t \right )-31 z \left (t \right )] \]

\[ {}[x^{\prime }\left (t \right ) = 6 x \left (t \right )-72 y \left (t \right )+44 z \left (t \right ), y^{\prime }\left (t \right ) = 4 x \left (t \right )-4 y \left (t \right )+26 z \left (t \right ), z^{\prime }\left (t \right ) = 6 x \left (t \right )-63 y \left (t \right )+38 z \left (t \right )] \]

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

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

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

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

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

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

\[ {}[x^{\prime }\left (t \right ) = x \left (t \right ) \left (a \left (p x \left (t \right )+q y \left (t \right )\right )+\alpha \right ), y^{\prime }\left (t \right ) = y \left (t \right ) \left (\beta +b \left (p x \left (t \right )+q y \left (t \right )\right )\right )] \]

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

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

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

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

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

\[ {}\left [x^{\prime }\left (t \right ) = -y \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ), y^{\prime }\left (t \right ) = \left \{\begin {array}{cc} x \left (t \right )^{2}+y \left (t \right )^{2} & 2 x \left (t \right )\le x \left (t \right )^{2}+y \left (t \right )^{2} \\ \left (\frac {x \left (t \right )}{2}-\frac {y \left (t \right )^{2}}{2 x \left (t \right )}\right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ) & \operatorname {otherwise} \end {array}\right .\right ] \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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