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

\[ {}3 y^{\prime \prime }+5 y^{\prime }-2 y = 7 \,{\mathrm e}^{-2 t} \]

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

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

\[ {}y^{\prime \prime }+9 y = 24 \sin \left (t \right ) \left (\operatorname {Heaviside}\left (t \right )+\operatorname {Heaviside}\left (-\pi +t \right )\right ) \]

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

\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 5 \cos \left (t \right ) \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \]

\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = 36 t \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right )\right ) \]

\[ {}y^{\prime \prime }+4 y^{\prime }+13 y = 39 \operatorname {Heaviside}\left (t \right )-507 \left (t -2\right ) \operatorname {Heaviside}\left (t -2\right ) \]

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

\[ {}4 y^{\prime \prime }+4 y^{\prime }+5 y = 25 t \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \]

\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = \operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right )+\operatorname {Heaviside}\left (t -2\right )-\operatorname {Heaviside}\left (-3+t \right ) \]

\[ {}y^{\prime \prime }-2 y^{\prime } = \left \{\begin {array}{cc} 4 & 0\le t <1 \\ 6 & 1\le t \end {array}\right . \]

\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \left \{\begin {array}{cc} 0 & 0\le t <1 \\ 1 & 1\le t <2 \\ -1 & 2\le t \end {array}\right . \]

\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \left \{\begin {array}{cc} 1 & 0\le t <2 \\ -1 & 2\le t \end {array}\right . \]

\[ {}y^{\prime \prime }+y = \left \{\begin {array}{cc} t & 0\le t <\pi \\ -t & \pi \le t \end {array}\right . \]

\[ {}y^{\prime \prime }+4 y = \left \{\begin {array}{cc} 8 t & 0\le t <\frac {\pi }{2} \\ 8 \pi & \frac {\pi }{2}\le t \end {array}\right . \]

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

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

\[ {}y^{\prime \prime }+4 y^{\prime }+29 y = 5 \left (\delta \left (-\pi +t \right )\right )-5 \left (\delta \left (-2 \pi +t \right )\right ) \]

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

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

\[ {}y^{\prime \prime }-7 y^{\prime }+6 y = \delta \left (t -1\right ) \]

\[ {}10 Q^{\prime }+100 Q = \operatorname {Heaviside}\left (t -1\right )-\operatorname {Heaviside}\left (t -2\right ) \]

\[ {}y^{\prime \prime \prime }+y^{\prime \prime }+4 y^{\prime }+4 y = 8 \]

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

\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+4 y^{\prime }-4 y = 8 \,{\mathrm e}^{2 t}-5 \,{\mathrm e}^{t} \]

\[ {}y^{\prime \prime \prime }-5 y^{\prime \prime }+y^{\prime }-y = -t^{2}+2 t -10 \]

\[ {}y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y = 12 \operatorname {Heaviside}\left (t \right )-12 \operatorname {Heaviside}\left (t -1\right ) \]

\[ {}y^{\prime \prime \prime \prime }-16 y = 32 \operatorname {Heaviside}\left (t \right )-32 \operatorname {Heaviside}\left (-\pi +t \right ) \]

\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }+y = t^{7} \]

\[ {}t^{2} y^{\prime \prime }-6 t y^{\prime }+\sin \left (2 t \right ) y = \ln \left (t \right ) \]

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

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

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

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

\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = {\mathrm e}^{t} \]

\[ {}y^{\prime \prime }-3 y^{\prime }-7 y = 4 \]

\[ {}y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y = 5 \]

\[ {}3 y^{\prime \prime }+5 y^{\prime }-2 y = 3 t^{2} \]

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

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

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

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

\[ {}[x^{\prime }\left (t \right )+5 x \left (t \right )-2 y \left (t \right ) = 0, y^{\prime }\left (t \right )+2 x \left (t \right )-y \left (t \right ) = 0] \]

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

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

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

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

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

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

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

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

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

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

\[ {}[x^{\prime }\left (t \right ) = -4 x \left (t \right )+9 y \left (t \right )+12 \,{\mathrm e}^{-t}, y^{\prime }\left (t \right ) = -5 x \left (t \right )+2 y \left (t \right )] \]

\[ {}[x^{\prime }\left (t \right ) = -7 x \left (t \right )+6 y \left (t \right )+6 \,{\mathrm e}^{-t}, y^{\prime }\left (t \right ) = -12 x \left (t \right )+5 y \left (t \right )+37] \]

\[ {}[x^{\prime }\left (t \right ) = -7 x \left (t \right )+10 y \left (t \right )+18 \,{\mathrm e}^{t}, y^{\prime }\left (t \right ) = -10 x \left (t \right )+9 y \left (t \right )+37] \]

\[ {}[x^{\prime }\left (t \right ) = -14 x \left (t \right )+39 y \left (t \right )+78 \sinh \left (t \right ), y^{\prime }\left (t \right ) = -6 x \left (t \right )+16 y \left (t \right )+6 \cosh \left (t \right )] \]

\[ {}[x^{\prime }\left (t \right ) = 2 x \left (t \right )+4 y \left (t \right )-2 z \left (t \right )-2 \sinh \left (t \right ), y^{\prime }\left (t \right ) = 4 x \left (t \right )+2 y \left (t \right )-2 z \left (t \right )+10 \cosh \left (t \right ), z^{\prime }\left (t \right ) = -x \left (t \right )+3 y \left (t \right )+z \left (t \right )+5] \]

\[ {}[x^{\prime }\left (t \right ) = 2 x \left (t \right )+6 y \left (t \right )-2 z \left (t \right )+50 \,{\mathrm e}^{t}, y^{\prime }\left (t \right ) = 6 x \left (t \right )+2 y \left (t \right )-2 z \left (t \right )+21 \,{\mathrm e}^{-t}, z^{\prime }\left (t \right ) = -x \left (t \right )+6 y \left (t \right )+z \left (t \right )+9] \]

\[ {}[x^{\prime }\left (t \right ) = -2 x \left (t \right )-2 y \left (t \right )+4 z \left (t \right ), y^{\prime }\left (t \right ) = -2 x \left (t \right )+y \left (t \right )+2 z \left (t \right ), z^{\prime }\left (t \right ) = -4 x \left (t \right )-2 y \left (t \right )+6 z \left (t \right )+{\mathrm e}^{2 t}] \]

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

\[ {}[x^{\prime }\left (t \right ) = 7 x \left (t \right )+y \left (t \right )-1-6 \,{\mathrm e}^{t}, y^{\prime }\left (t \right ) = -4 x \left (t \right )+3 y \left (t \right )+4 \,{\mathrm e}^{t}-3] \]

\[ {}[x^{\prime }\left (t \right ) = 3 x \left (t \right )-2 y \left (t \right )+24 \sin \left (t \right ), y^{\prime }\left (t \right ) = 9 x \left (t \right )-3 y \left (t \right )+12 \cos \left (t \right )] \]

\[ {}[x^{\prime }\left (t \right ) = 7 x \left (t \right )-4 y \left (t \right )+10 \,{\mathrm e}^{t}, y^{\prime }\left (t \right ) = 3 x \left (t \right )+14 y \left (t \right )+6 \,{\mathrm e}^{2 t}] \]

\[ {}[x^{\prime }\left (t \right ) = -7 x \left (t \right )+4 y \left (t \right )+6 \,{\mathrm e}^{3 t}, y^{\prime }\left (t \right ) = -5 x \left (t \right )+2 y \left (t \right )+6 \,{\mathrm e}^{2 t}] \]

\[ {}[x^{\prime }\left (t \right ) = -3 x \left (t \right )-3 y \left (t \right )+z \left (t \right ), y^{\prime }\left (t \right ) = 2 y \left (t \right )+2 z \left (t \right )+29 \,{\mathrm e}^{-t}, z^{\prime }\left (t \right ) = 5 x \left (t \right )+y \left (t \right )+z \left (t \right )+39 \,{\mathrm e}^{t}] \]

\[ {}[x^{\prime }\left (t \right ) = 2 x \left (t \right )+y \left (t \right )-z \left (t \right )+5 \sin \left (t \right ), y^{\prime }\left (t \right ) = y \left (t \right )+z \left (t \right )-10 \cos \left (t \right ), z^{\prime }\left (t \right ) = x \left (t \right )+z \left (t \right )+2] \]

\[ {}[x^{\prime }\left (t \right ) = -3 x \left (t \right )+3 y \left (t \right )+z \left (t \right )+5 \sin \left (2 t \right ), y^{\prime }\left (t \right ) = x \left (t \right )-5 y \left (t \right )-3 z \left (t \right )+5 \cos \left (2 t \right ), z^{\prime }\left (t \right ) = -3 x \left (t \right )+7 y \left (t \right )+3 z \left (t \right )+23 \,{\mathrm e}^{t}] \]

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

\[ {}[x^{\prime }\left (t \right ) = x \left (t \right )+5 y \left (t \right )+10 \sinh \left (t \right ), y^{\prime }\left (t \right ) = 19 x \left (t \right )-13 y \left (t \right )+24 \sinh \left (t \right )] \]

\[ {}[x^{\prime }\left (t \right ) = 9 x \left (t \right )-3 y \left (t \right )-6 t, y^{\prime }\left (t \right ) = -x \left (t \right )+11 y \left (t \right )+10 t] \]

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

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

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

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

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

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

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

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

\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (1-5 x \right ) y^{\prime }-4 y = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime \prime }+\beta y^{\prime }+\gamma y = 0 \]

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

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

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

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

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