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

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

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

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

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

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

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

\[ {}x y^{\prime \prime }-3 y^{\prime } = 5 x \]

\[ {}y^{\prime \prime }+3 y^{\prime }-10 y = 6 \,{\mathrm e}^{4 x} \]

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

\[ {}y^{\prime \prime }+10 y^{\prime }+25 y = 14 \,{\mathrm e}^{-5 x} \]

\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 25 x^{2}+12 \]

\[ {}y^{\prime \prime }-y^{\prime }-6 y = 20 \,{\mathrm e}^{-2 x} \]

\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 14 \sin \left (2 x \right )-18 \cos \left (2 x \right ) \]

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

\[ {}y^{\prime \prime }-2 y^{\prime } = 12 x -10 \]

\[ {}y^{\prime \prime }-2 y^{\prime }+y = 6 \,{\mathrm e}^{x} \]

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

\[ {}y^{\prime \prime }+y^{\prime } = 10 x^{4}+2 \]

\[ {}y^{\prime \prime }+4 y = 4 \cos \left (2 x \right )+6 \cos \left (x \right )+8 x^{2}-4 x \]

\[ {}y^{\prime \prime }+9 y = 2 \sin \left (3 x \right )+4 \sin \left (x \right )-26 \,{\mathrm e}^{-2 x}+27 x^{3} \]

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

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

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

\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 64 x \,{\mathrm e}^{-x} \]

\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = {\mathrm e}^{-x} \sec \left (2 x \right ) \]

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

\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \frac {1}{1+{\mathrm e}^{-x}} \]

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

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

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

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

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

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

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

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

\[ {}y^{\prime \prime }-y^{\prime }-6 y = {\mathrm e}^{-x} \]

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

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

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

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

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

\[ {}y^{\prime \prime }-2 y^{\prime }-5 y = x \]

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 2 x -1 \]

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

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

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

\[ {}y^{\prime \prime }+9 y = \sec \left (2 x \right ) \]

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

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

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

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

\[ {}y^{\prime \prime }+y = -8 \sin \left (3 x \right ) \]

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

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

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

\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = 5 \,{\mathrm e}^{3 t} \]

\[ {}y^{\prime \prime }+y^{\prime }-6 y = t \]

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

\[ {}y^{\prime \prime }+3 y^{\prime }-5 y = 1 \]

\[ {}y^{\prime \prime }+3 y^{\prime }-2 y = -6 \,{\mathrm e}^{\pi -t} \]

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

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

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

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

\[ {}y^{\prime \prime }-7 y^{\prime }+12 y = t \,{\mathrm e}^{2 t} \]

\[ {}i^{\prime \prime }+2 i^{\prime }+3 i = \left \{\begin {array}{cc} 30 & 0<t <2 \pi \\ 0 & 2 \pi \le t \le 5 \pi \\ 10 & 5 \pi <t <\infty \end {array}\right . \]

\[ {}y^{\prime \prime }-4 y^{\prime } = 6 \,{\mathrm e}^{3 t}-3 \,{\mathrm e}^{-t} \]

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

\[ {}y^{\prime \prime }+9 y = {\mathrm e}^{t} \]

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

\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = t \]

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

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

\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = t +1 \]

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

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

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

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

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

\[ {}y^{\prime \prime }+9 y = \cos \left (3 t \right ) \]

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

\[ {}y^{\prime \prime }+16 y = \left \{\begin {array}{cc} \cos \left (4 t \right ) & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \]

\[ {}y^{\prime \prime }+y = \left \{\begin {array}{cc} 1 & 0\le t <\frac {\pi }{2} \\ \sin \left (t \right ) & \frac {\pi }{2}\le t \end {array}\right . \]

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

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

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

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

\[ {}y^{\prime \prime }+16 y = \delta \left (t -2 \pi \right ) \]

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