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

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

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

\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+4 x^{4} y = 0 \]

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

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

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

\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \]

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

\[ {}y^{\prime \prime } = \frac {6 y}{x^{2}} \]

\[ {}y^{\prime \prime } = \left (-\frac {3}{16 x^{2}}-\frac {2}{9 \left (-1+x \right )^{2}}+\frac {3}{16 x \left (-1+x \right )}\right ) y \]

\[ {}y^{\prime \prime } = \frac {20 y}{x^{2}} \]

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

\[ {}y^{\prime \prime }-\frac {y}{4 x^{2}} = 0 \]

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

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

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

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

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

\[ {}y^{\prime \prime } = \frac {\left (4 x^{6}-8 x^{5}+12 x^{4}+4 x^{3}+7 x^{2}-20 x +4\right ) y}{4 x^{4}} \]

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

\[ {}y^{\prime \prime } = \left (\frac {x^{2}}{4}-\frac {11}{2}\right ) y \]

\[ {}y^{\prime \prime } = \left (\frac {1}{x}-\frac {3}{16 x^{2}}\right ) y \]

\[ {}y^{\prime \prime } = \left (-\frac {3}{16 x^{2}}-\frac {2}{9 \left (-1+x \right )^{2}}+\frac {3}{16 x \left (-1+x \right )}\right ) y \]

\[ {}y^{\prime \prime } = -\frac {\left (5 x^{2}+27\right ) y}{36 \left (x^{2}-1\right )^{2}} \]

\[ {}y^{\prime \prime } = -\frac {y}{4 x^{2}} \]

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime \prime }+a \,{\mathrm e}^{b x} y = 0 \]

\[ {}y^{\prime \prime }-\left (4 a^{2} b^{2} x^{2} {\mathrm e}^{2 b \,x^{2}}-1\right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{x}+c \right ) y = 0 \]

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

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

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

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

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

\[ {}y^{\prime \prime }-\left (n \left (n +1\right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+B \right ) y = 0 \]

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

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

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

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

\[ {}y^{\prime \prime }+y^{\prime }+a \,{\mathrm e}^{-2 x} y = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (3 x^{2}+2 n -1\right ) y = 0 \]

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

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

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

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y = 0 \]

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

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

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

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

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

\[ {}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = 0 \]

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

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

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

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

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

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

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

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

\[ {}y^{\prime \prime }+\frac {\left (11 \operatorname {WeierstrassP}\left (x , a , b\right ) \operatorname {WeierstrassPPrime}\left (x , a , b\right )-6 \operatorname {WeierstrassP}\left (x , a , b\right )^{2}+\frac {a}{2}\right ) y^{\prime }}{\operatorname {WeierstrassPPrime}\left (x , a , b\right )+\operatorname {WeierstrassP}\left (x , a , b\right )^{2}}+\frac {\left (\operatorname {WeierstrassPPrime}\left (x , a , b\right )^{2}-\operatorname {WeierstrassP}\left (x , a , b\right )^{2} \operatorname {WeierstrassPPrime}\left (x , a , b\right )-\operatorname {WeierstrassP}\left (x , a , b\right ) \left (6 \operatorname {WeierstrassP}\left (x , a , b\right )^{2}-\frac {a}{2}\right )\right ) y}{\operatorname {WeierstrassPPrime}\left (x , a , b\right )+\operatorname {WeierstrassP}\left (x , a , b\right )^{2}} = 0 \]

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

\[ {}y^{\prime \prime }+\left (a f \left (x \right )+b \right ) y^{\prime }+\left (c f \left (x \right )+d \right ) y = 0 \]

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

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

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

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

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

\[ {}4 y^{\prime \prime }+9 x y = 0 \]

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

\[ {}4 y^{\prime \prime }+4 \tan \left (x \right ) y^{\prime }-\left (5 \tan \left (x \right )^{2}+2\right ) y = 0 \]

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

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

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

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