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

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

\[ {}4 t y^{\prime \prime }+3 y^{\prime }-3 y = 0 \]

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

\[ {}t^{3} y^{\prime \prime }-t y^{\prime }-\left (t^{2}+\frac {5}{4}\right ) y = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}[x_{1}^{\prime }\left (t \right ) = 3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+4 x_{3} \left (t \right ), x_{2}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )+2 x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = 4 x_{1} \left (t \right )+2 x_{2} \left (t \right )+3 x_{3} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = 7 x_{1} \left (t \right )-x_{2} \left (t \right )+6 x_{3} \left (t \right ), x_{2}^{\prime }\left (t \right ) = -10 x_{1} \left (t \right )+4 x_{2} \left (t \right )-12 x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = -2 x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = -7 x_{1} \left (t \right )+6 x_{3} \left (t \right ), x_{2}^{\prime }\left (t \right ) = 5 x_{2} \left (t \right ), x_{3}^{\prime }\left (t \right ) = 6 x_{1} \left (t \right )+2 x_{3} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right )+2 x_{2} \left (t \right )+3 x_{3} \left (t \right )+6 x_{4} \left (t \right ), x_{2}^{\prime }\left (t \right ) = 3 x_{1} \left (t \right )+6 x_{2} \left (t \right )+9 x_{3} \left (t \right )+18 x_{4} \left (t \right ), x_{3}^{\prime }\left (t \right ) = 5 x_{1} \left (t \right )+10 x_{2} \left (t \right )+15 x_{3} \left (t \right )+30 x_{4} \left (t \right ), x_{4}^{\prime }\left (t \right ) = 7 x_{1} \left (t \right )+14 x_{2} \left (t \right )+21 x_{3} \left (t \right )+42 x_{4} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right )+x_{2} \left (t \right ), x_{2}^{\prime }\left (t \right ) = 4 x_{1} \left (t \right )+x_{2} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right )-3 x_{2} \left (t \right ), x_{2}^{\prime }\left (t \right ) = -2 x_{1} \left (t \right )+2 x_{2} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = 3 x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right ), x_{2}^{\prime }\left (t \right ) = x_{1} \left (t \right )+3 x_{2} \left (t \right )-x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = 3 x_{1} \left (t \right )+3 x_{2} \left (t \right )-x_{3} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right )-x_{2} \left (t \right ), x_{2}^{\prime }\left (t \right ) = x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = x_{1} \left (t \right )+10 x_{2} \left (t \right )+2 x_{3} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right )-3 x_{2} \left (t \right )+2 x_{3} \left (t \right ), x_{2}^{\prime }\left (t \right ) = -x_{2} \left (t \right ), x_{3}^{\prime }\left (t \right ) = -x_{2} \left (t \right )-2 x_{3} \left (t \right )] \]

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

\[ {}[x_{1}^{\prime }\left (t \right ) = -3 x_{1} \left (t \right )+2 x_{2} \left (t \right ), x_{2}^{\prime }\left (t \right ) = -x_{1} \left (t \right )-x_{2} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right )-5 x_{2} \left (t \right ), x_{2}^{\prime }\left (t \right ) = x_{1} \left (t \right )-3 x_{2} \left (t \right ), x_{3}^{\prime }\left (t \right ) = x_{3} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right ), x_{2}^{\prime }\left (t \right ) = 3 x_{1} \left (t \right )+x_{2} \left (t \right )-2 x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right )+x_{3} \left (t \right ), x_{2}^{\prime }\left (t \right ) = x_{2} \left (t \right )-x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = -2 x_{1} \left (t \right )-x_{3} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right )-x_{2} \left (t \right ), x_{2}^{\prime }\left (t \right ) = 5 x_{1} \left (t \right )-3 x_{2} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = 3 x_{1} \left (t \right )-2 x_{2} \left (t \right ), x_{2}^{\prime }\left (t \right ) = 4 x_{1} \left (t \right )-x_{2} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = -3 x_{1} \left (t \right )+2 x_{3} \left (t \right ), x_{2}^{\prime }\left (t \right ) = x_{1} \left (t \right )-x_{2} \left (t \right ), x_{3}^{\prime }\left (t \right ) = -2 x_{1} \left (t \right )-x_{2} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = 2 x_{2} \left (t \right ), x_{2}^{\prime }\left (t \right ) = -2 x_{1} \left (t \right ), x_{3}^{\prime }\left (t \right ) = -3 x_{4} \left (t \right ), x_{4}^{\prime }\left (t \right ) = 3 x_{3} \left (t \right )] \]

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

\[ {}[x_{1}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )+x_{2} \left (t \right )+3 x_{3} \left (t \right ), x_{2}^{\prime }\left (t \right ) = 2 x_{2} \left (t \right )-x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = 2 x_{3} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = -x_{2} \left (t \right )+x_{3} \left (t \right ), x_{2}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )-3 x_{2} \left (t \right )+x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = x_{1} \left (t \right )-x_{2} \left (t \right )-x_{3} \left (t \right )] \]

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

\[ {}[x_{1}^{\prime }\left (t \right ) = -x_{1} \left (t \right )-x_{2} \left (t \right ), x_{2}^{\prime }\left (t \right ) = -x_{2} \left (t \right ), x_{3}^{\prime }\left (t \right ) = -2 x_{3} \left (t \right )] \]

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

\[ {}[x_{1}^{\prime }\left (t \right ) = -x_{1} \left (t \right )+x_{2} \left (t \right )+2 x_{3} \left (t \right ), x_{2}^{\prime }\left (t \right ) = -x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = -2 x_{1} \left (t \right )+x_{2} \left (t \right )+3 x_{3} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = -4 x_{1} \left (t \right )-4 x_{2} \left (t \right ), x_{2}^{\prime }\left (t \right ) = 10 x_{1} \left (t \right )+9 x_{2} \left (t \right )+x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = -4 x_{1} \left (t \right )-3 x_{2} \left (t \right )+x_{3} \left (t \right )] \]

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

\[ {}[x_{1}^{\prime }\left (t \right ) = 3 x_{1} \left (t \right ), x_{2}^{\prime }\left (t \right ) = x_{1} \left (t \right )+3 x_{2} \left (t \right ), x_{3}^{\prime }\left (t \right ) = 3 x_{3} \left (t \right ), x_{4}^{\prime }\left (t \right ) = 2 x_{3} \left (t \right )+3 x_{4} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right ), x_{2}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )+x_{2} \left (t \right )-2 x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = 3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right )+{\mathrm e}^{t} \cos \left (2 t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right )+{\mathrm e}^{c t}, x_{2}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )+x_{2} \left (t \right )-2 x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = 3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = 4 x_{1} \left (t \right )+5 x_{2} \left (t \right )+4 \,{\mathrm e}^{t} \cos \left (t \right ), x_{2}^{\prime }\left (t \right ) = -2 x_{1} \left (t \right )-2 x_{2} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = 3 x_{1} \left (t \right )-4 x_{2} \left (t \right )+{\mathrm e}^{t}, x_{2}^{\prime }\left (t \right ) = x_{1} \left (t \right )-x_{2} \left (t \right )+{\mathrm e}^{t}] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )-5 x_{2} \left (t \right )+\sin \left (t \right ), x_{2}^{\prime }\left (t \right ) = x_{1} \left (t \right )-2 x_{2} \left (t \right )+\tan \left (t \right )] \]

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

\[ {}[x_{1}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )+x_{3} \left (t \right )+{\mathrm e}^{2 t}, x_{2}^{\prime }\left (t \right ) = 2 x_{2} \left (t \right ), x_{3}^{\prime }\left (t \right ) = x_{2} \left (t \right )+3 x_{3} \left (t \right )+{\mathrm e}^{2 t}] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = -x_{1} \left (t \right )-x_{2} \left (t \right )-2 x_{3} \left (t \right )+{\mathrm e}^{t}, x_{2}^{\prime }\left (t \right ) = x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )+x_{2} \left (t \right )+3 x_{3} \left (t \right )] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )+x_{2} \left (t \right )+{\mathrm e}^{3 t}, x_{2}^{\prime }\left (t \right ) = 3 x_{1} \left (t \right )-2 x_{2} \left (t \right )+{\mathrm e}^{3 t}] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right )-x_{2} \left (t \right )-t^{2}, x_{2}^{\prime }\left (t \right ) = x_{1} \left (t \right )+3 x_{2} \left (t \right )+2 t] \]

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

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right )+2 x_{2} \left (t \right )-3 x_{3} \left (t \right )+{\mathrm e}^{t}, x_{2}^{\prime }\left (t \right ) = x_{1} \left (t \right )+x_{2} \left (t \right )+2 x_{3} \left (t \right ), x_{3}^{\prime }\left (t \right ) = x_{1} \left (t \right )-x_{2} \left (t \right )+4 x_{3} \left (t \right )-{\mathrm e}^{t}] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = -x_{1} \left (t \right )-x_{2} \left (t \right )+1, x_{2}^{\prime }\left (t \right ) = -4 x_{2} \left (t \right )-x_{3} \left (t \right )+t, x_{3}^{\prime }\left (t \right ) = 5 x_{2} \left (t \right )+{\mathrm e}^{t}] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )+{\mathrm e}^{2 t}, x_{2}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )+3 x_{2} \left (t \right )-4 x_{3} \left (t \right )+2 \,{\mathrm e}^{2 t}, x_{3}^{\prime }\left (t \right ) = 4 x_{1} \left (t \right )+x_{2} \left (t \right )-4 x_{3} \left (t \right )+{\mathrm e}^{2 t}] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = x_{1} \left (t \right )-x_{2} \left (t \right )-x_{3} \left (t \right )+{\mathrm e}^{3 t}, x_{2}^{\prime }\left (t \right ) = x_{1} \left (t \right )+3 x_{2} \left (t \right )+x_{3} \left (t \right )-{\mathrm e}^{3 t}, x_{3}^{\prime }\left (t \right ) = -3 x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )-{\mathrm e}^{3 t}] \]

\[ {}[x_{1}^{\prime }\left (t \right ) = 3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+4 x_{3} \left (t \right )+2 \,{\mathrm e}^{8 t}, x_{2}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )+2 x_{3} \left (t \right )+{\mathrm e}^{8 t}, x_{3}^{\prime }\left (t \right ) = 4 x_{1} \left (t \right )+2 x_{2} \left (t \right )+3 x_{3} \left (t \right )+2 \,{\mathrm e}^{8 t}] \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}\left (x^{2}-2 x -8\right ) y^{\prime } = -2+y^{2}+y \]

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