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

\[ {}t^{2} y^{\prime \prime }+7 t y^{\prime }+5 y = t \]

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

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

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

\[ {}u^{\prime \prime }+\frac {u^{\prime }}{4}+2 u = 0 \]

\[ {}u^{\prime \prime }+\frac {u^{\prime }}{8}+4 u = 3 \cos \left (\frac {t}{4}\right ) \]

\[ {}u^{\prime \prime }+\frac {u^{\prime }}{8}+4 u = 3 \cos \left (2 t \right ) \]

\[ {}u^{\prime \prime }+\frac {u^{\prime }}{8}+4 u = 3 \cos \left (6 t \right ) \]

\[ {}u^{\prime \prime }+u^{\prime }+\frac {u^{3}}{5} = \cos \left (t \right ) \]

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

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

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

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

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

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

\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime \prime }+4 y^{\prime }+6 x y = 0 \]

\[ {}y^{\prime \prime }+4 y^{\prime }+6 x y = 0 \]

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

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

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

\[ {}\left (x^{3}+1\right ) y^{\prime \prime }+4 x y^{\prime }+y = 0 \]

\[ {}\left (x^{3}+1\right ) y^{\prime \prime }+4 x y^{\prime }+y = 0 \]

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

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

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

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

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

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

\[ {}\left [x_{1}^{\prime }\left (t \right ) = -\frac {x_{1} \left (t \right )}{10}+\frac {3 x_{2} \left (t \right )}{40}, x_{2}^{\prime }\left (t \right ) = \frac {x_{1} \left (t \right )}{10}-\frac {x_{2} \left (t \right )}{5}\right ] \]

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

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

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

\[ {}\left [x_{1}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )-\frac {5 x_{2} \left (t \right )}{2}, x_{2}^{\prime }\left (t \right ) = \frac {9 x_{1} \left (t \right )}{5}-x_{2} \left (t \right )\right ] \]

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

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

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

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

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

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

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

\[ {}\left [x_{1}^{\prime }\left (t \right ) = -\frac {4 x_{1} \left (t \right )}{5}+2 x_{2} \left (t \right ), x_{2}^{\prime }\left (t \right ) = -x_{1} \left (t \right )+\frac {6 x_{2} \left (t \right )}{5}\right ] \]

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

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

\[ {}\left [x_{1}^{\prime }\left (t \right ) = -\frac {x_{1} \left (t \right )}{2}-\frac {x_{2} \left (t \right )}{8}, x_{2}^{\prime }\left (t \right ) = 2 x_{1} \left (t \right )-\frac {x_{2} \left (t \right )}{2}\right ] \]

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

\[ {}[x_{1}^{\prime }\left (t \right ) = 4 x_{1} \left (t \right )-2 x_{2} \left (t \right ), x_{2}^{\prime }\left (t \right ) = 8 x_{1} \left (t \right )-4 x_{2} \left (t \right )] \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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