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

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

\[ {}y^{\prime \prime } = \frac {\left (5 x -4\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (9 x -6\right ) y}{x^{2} \left (-1+x \right )} \]

\[ {}y^{\prime \prime } = -\frac {\left (\left (a +b +1\right ) x +\alpha +\beta -1\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (-1+x \right )} \]

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

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

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

\[ {}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +\delta \right )-\delta \right ) x +a \gamma \right ) y^{\prime }}{x \left (-1+x \right ) \left (x -a \right )}-\frac {\left (\alpha \beta x -q \right ) y}{x \left (-1+x \right ) \left (x -a \right )} \]

\[ {}y^{\prime \prime } = -\frac {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}-\frac {\left (\mathit {DD} x +E \right ) y}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )} \]

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

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

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

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

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

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

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

\[ {}y^{\prime \prime } = \frac {\left (6 x -1\right ) y^{\prime }}{3 x \left (-2+x \right )}+\frac {y}{3 x^{2} \left (-2+x \right )} \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime \prime } = \frac {12 y}{\left (1+x \right )^{2} \left (x^{2}+2 x +3\right )} \]

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

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

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

\[ {}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) \left (x -a \right )^{2} \left (x -b \right )+\left (1-\alpha -\beta \right ) \left (x -b \right )^{2} \left (x -a \right )\right ) y^{\prime }}{\left (x -a \right )^{2} \left (x -b \right )^{2}}-\frac {\alpha \beta \left (a -b \right )^{2} y}{\left (x -a \right )^{2} \left (x -b \right )^{2}} \]

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

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

\[ {}y^{\prime \prime } = \frac {18 y}{\left (1+2 x \right )^{2} \left (x^{2}+x +1\right )} \]

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

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

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

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

\[ {}y^{\prime \prime } = \frac {\left (7 a \,x^{2}+5\right ) y^{\prime }}{x \left (a \,x^{2}+1\right )}-\frac {\left (15 a \,x^{2}+5\right ) y}{x^{2} \left (a \,x^{2}+1\right )} \]

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

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

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

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

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

\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x^{4}}+\frac {y}{x^{5}} \]

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

\[ {}y^{\prime \prime } = \frac {\left (3 x +1\right ) y^{\prime }}{\left (-1+x \right ) \left (1+x \right )}-\frac {36 \left (1+x \right )^{2} y}{\left (-1+x \right )^{2} \left (3 x +5\right )^{2}} \]

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

\[ {}y^{\prime \prime } = -\frac {\left (3 x^{2}+a \right ) y^{\prime }}{x^{3}}-\frac {b y}{x^{6}} \]

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

\[ {}y^{\prime \prime } = -\left (\frac {1-\mathit {a1} -\mathit {b1}}{x -\mathit {c1}}+\frac {1-\mathit {a2} -\mathit {b2}}{x -\mathit {c2}}+\frac {1-\mathit {a3} -\mathit {b3}}{x -\mathit {c3}}\right ) y^{\prime }-\frac {\left (\frac {\mathit {a1} \mathit {b1} \left (\mathit {c1} -\mathit {c3} \right ) \left (\mathit {c1} -\mathit {c2} \right )}{x -\mathit {c1}}+\frac {\mathit {a2} \mathit {b2} \left (\mathit {c2} -\mathit {c1} \right ) \left (\mathit {c2} -\mathit {c3} \right )}{x -\mathit {c2}}+\frac {\mathit {a3} \mathit {b3} \left (\mathit {c3} -\mathit {c2} \right ) \left (\mathit {c3} -\mathit {c1} \right )}{x -\mathit {c3}}\right ) y}{\left (x -\mathit {c1} \right ) \left (x -\mathit {c2} \right ) \left (x -\mathit {c3} \right )} \]

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

\[ {}y^{\prime \prime } = \frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (a \,x^{4}+10 x^{2}+1\right ) y}{4 x^{6}} \]

\[ {}y^{\prime \prime } = -\frac {27 x y}{16 \left (x^{3}-1\right )^{2}} \]

\[ {}y^{\prime \prime } = -\left (\frac {\left (1-\mathit {al1} -\mathit {bl1} \right ) \mathit {b1}}{\mathit {b1} x -\mathit {a1}}+\frac {\left (1-\mathit {al2} -\mathit {bl2} \right ) \mathit {b2}}{\mathit {b2} x -\mathit {a2}}+\frac {\left (1-\mathit {al3} -\mathit {bl3} \right ) \mathit {b3}}{\mathit {b3} x -\mathit {a3}}\right ) y^{\prime }-\frac {\left (\frac {\mathit {al1} \mathit {bl1} \left (\mathit {a1} \mathit {b2} -\mathit {a2} \mathit {b1} \right ) \left (-\mathit {a1} \mathit {b3} +\mathit {a3} \mathit {b1} \right )}{\mathit {b1} x -\mathit {a1}}+\frac {\mathit {al2} \mathit {bl2} \left (\mathit {a2} \mathit {b3} -\mathit {a3} \mathit {b2} \right ) \left (\mathit {a1} \mathit {b2} -\mathit {a2} \mathit {b1} \right )}{\mathit {b2} x -\mathit {a2}}+\frac {\mathit {al3} \mathit {bl3} \left (-\mathit {a1} \mathit {b3} +\mathit {a3} \mathit {b1} \right ) \left (\mathit {a2} \mathit {b3} -\mathit {a3} \mathit {b2} \right )}{\mathit {b3} x -\mathit {a3}}\right ) y}{\left (\mathit {b1} x -\mathit {a1} \right ) \left (\mathit {b2} x -\mathit {a2} \right ) \left (\mathit {b3} x -\mathit {a3} \right )} \]

\[ {}y^{\prime \prime } = -\frac {\left (x^{2} \left (\left (x^{2}-\mathit {a1} \right ) \left (x^{2}-\mathit {a2} \right )+\left (x^{2}-\mathit {a2} \right ) \left (x^{2}-\mathit {a3} \right )+\left (x^{2}-\mathit {a3} \right ) \left (x^{2}-\mathit {a1} \right )\right )-\left (x^{2}-\mathit {a1} \right ) \left (x^{2}-\mathit {a2} \right ) \left (x^{2}-\mathit {a3} \right )\right ) y^{\prime }}{x \left (x^{2}-\mathit {a1} \right ) \left (x^{2}-\mathit {a2} \right ) \left (x^{2}-\mathit {a3} \right )}-\frac {\left (A \,x^{2}+B \right ) y}{x \left (x^{2}-\mathit {a1} \right ) \left (x^{2}-\mathit {a2} \right ) \left (x^{2}-\mathit {a3} \right )} \]

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

\[ {}y^{\prime \prime } = -\frac {\left (a p \,x^{b}+q \right ) y^{\prime }}{x \left (a \,x^{b}-1\right )}-\frac {\left (a r \,x^{b}+s \right ) y}{x^{2} \left (a \,x^{b}-1\right )} \]

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

\[ {}y^{\prime \prime } = \frac {y^{\prime }}{x \ln \relax (x )}+\ln \relax (x )^{2} y \]

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

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

\[ {}y^{\prime \prime } = -\frac {2 n \cosh \relax (x ) y^{\prime }}{\sinh \relax (x )}-\left (-a^{2}+n^{2}\right ) y \]

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

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

\[ {}y^{\prime \prime } = -\frac {x \sin \relax (x ) y^{\prime }}{\cos \relax (x ) x -\sin \relax (x )}+\frac {\sin \relax (x ) y}{\cos \relax (x ) x -\sin \relax (x )} \]

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

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

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