3.392 problem 1398

Internal problem ID [9725]
Internal file name [OUTPUT/8667_Monday_June_06_2022_04_44_47_AM_52059280/index.tex]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1398.
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "unknown"

Maple gives the following as the ode type

[[_2nd_order, _with_linear_symmetries]]

Unable to solve or complete the solution.

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

3.392.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) \left (x -1\right )^{2} \left (x +1\right )^{2} x +\left (3 x^{4}-4 x^{2}+1\right ) y^{\prime }-\left (4 v^{2}-x^{2}+4 v +2\right ) y x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\frac {\left (4 v^{2}-x^{2}+4 v +2\right ) y}{\left (x -1\right )^{2} \left (x +1\right )^{2}}-\frac {\left (3 x^{2}-1\right ) y^{\prime }}{x \left (x -1\right ) \left (x +1\right )} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }+\frac {\left (3 x^{2}-1\right ) y^{\prime }}{x \left (x -1\right ) \left (x +1\right )}-\frac {\left (4 v^{2}-x^{2}+4 v +2\right ) y}{\left (x -1\right )^{2} \left (x +1\right )^{2}}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {3 x^{2}-1}{x \left (x -1\right ) \left (x +1\right )}, P_{3}\left (x \right )=-\frac {4 v^{2}-x^{2}+4 v +2}{\left (x -1\right )^{2} \left (x +1\right )^{2}}\right ] \\ {} & \circ & \left (x +1\right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-1 \\ {} & {} & \left (\left (x +1\right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-1}}}=1 \\ {} & \circ & \left (x +1\right )^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-1 \\ {} & {} & \left (\left (x +1\right )^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-1}}}=-v^{2}-v -\frac {1}{4} \\ {} & \circ & x =-1\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=-1 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) \left (x -1\right )^{2} \left (x +1\right )^{2} x +\left (x -1\right ) \left (x +1\right ) \left (3 x^{2}-1\right ) y^{\prime }-\left (4 v^{2}-x^{2}+4 v +2\right ) y x =0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u -1\hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & \left (u^{5}-5 u^{4}+8 u^{3}-4 u^{2}\right ) \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )+\left (3 u^{4}-12 u^{3}+14 u^{2}-4 u \right ) \left (\frac {d}{d u}y \left (u \right )\right )+\left (u^{3}-4 u \,v^{2}-3 u^{2}-4 u v +4 v^{2}+u +4 v +1\right ) y \left (u \right )=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \left (u \right ) \\ {} & {} & y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot y \left (u \right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..3 \\ {} & {} & u^{m}\cdot y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & u^{m}\cdot y \left (u \right )=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} u^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..4 \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) u^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) u^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =2..5 \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) u^{k +r -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-2+m}{\sum }}a_{k +2-m} \left (k +2-m +r \right ) \left (k +1-m +r \right ) u^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & -a_{0} \left (2 v +1+2 r \right ) \left (-2 v -1+2 r \right ) u^{r}+\left (-a_{1} \left (2 v +3+2 r \right ) \left (-2 v +1+2 r \right )+a_{0} \left (8 r^{2}-4 v^{2}+6 r -4 v +1\right )\right ) u^{1+r}+\left (-a_{2} \left (2 v +5+2 r \right ) \left (-2 v +3+2 r \right )+a_{1} \left (8 r^{2}-4 v^{2}+22 r -4 v +15\right )-a_{0} \left (5 r^{2}+7 r +3\right )\right ) u^{2+r}+\left (\moverset {\infty }{\munderset {k =3}{\sum }}\left (-a_{k} \left (2 v +1+2 r +2 k \right ) \left (-2 v -1+2 r +2 k \right )+a_{k -1} \left (8 \left (k -1\right )^{2}+16 \left (k -1\right ) r +8 r^{2}-4 v^{2}+6 k -5+6 r -4 v \right )-a_{k -2} \left (5 \left (k -2\right )^{2}+10 \left (k -2\right ) r +5 r^{2}+7 k -11+7 r \right )+a_{k -3} \left (k -2+r \right )^{2}\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -\left (2 v +1+2 r \right ) \left (-2 v -1+2 r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-v -\frac {1}{2}, v +\frac {1}{2}\right \} \\ \bullet & {} & \textrm {The coefficients of each power of}\hspace {3pt} u \hspace {3pt}\textrm {must be 0}\hspace {3pt} \\ {} & {} & \left [-a_{1} \left (2 v +3+2 r \right ) \left (-2 v +1+2 r \right )+a_{0} \left (8 r^{2}-4 v^{2}+6 r -4 v +1\right )=0, -a_{2} \left (2 v +5+2 r \right ) \left (-2 v +3+2 r \right )+a_{1} \left (8 r^{2}-4 v^{2}+22 r -4 v +15\right )-a_{0} \left (5 r^{2}+7 r +3\right )=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{1}=\frac {a_{0} \left (8 r^{2}-4 v^{2}+6 r -4 v +1\right )}{4 r^{2}-4 v^{2}+8 r -4 v +3}, a_{2}=\frac {a_{0} \left (44 r^{4}-44 r^{2} v^{2}+16 v^{4}+156 r^{3}-44 r^{2} v -84 v^{2} r +32 v^{3}+177 r^{2}-84 v r -36 v^{2}+67 r -52 v +6\right )}{16 r^{4}-32 r^{2} v^{2}+16 v^{4}+96 r^{3}-32 r^{2} v -96 v^{2} r +32 v^{3}+200 r^{2}-96 v r -56 v^{2}+168 r -72 v +45}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (-4 a_{k}+a_{k -3}-5 a_{k -2}+8 a_{k -1}\right ) r^{2}+\left (\left (-10 k +13\right ) a_{k -2}+2 \left (-4 a_{k}+a_{k -3}+8 a_{k -1}\right ) k -4 a_{k -3}-10 a_{k -1}\right ) r +\left (-5 k^{2}+13 k -9\right ) a_{k -2}+\left (-4 a_{k}+a_{k -3}+8 a_{k -1}\right ) k^{2}+2 \left (-2 a_{k -3}-5 a_{k -1}\right ) k +4 \left (v +\frac {1}{2}\right )^{2} a_{k}-4 v^{2} a_{k -1}-4 v a_{k -1}+4 a_{k -3}+3 a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +3 \\ {} & {} & \left (-4 a_{k +3}+a_{k}-5 a_{k +1}+8 a_{k +2}\right ) r^{2}+\left (\left (-10 k -17\right ) a_{k +1}+2 \left (-4 a_{k +3}+a_{k}+8 a_{k +2}\right ) \left (k +3\right )-4 a_{k}-10 a_{k +2}\right ) r +\left (-5 \left (k +3\right )^{2}+13 k +30\right ) a_{k +1}+\left (-4 a_{k +3}+a_{k}+8 a_{k +2}\right ) \left (k +3\right )^{2}+2 \left (-2 a_{k}-5 a_{k +2}\right ) \left (k +3\right )+4 \left (v +\frac {1}{2}\right )^{2} a_{k +3}-4 v^{2} a_{k +2}-4 v a_{k +2}+4 a_{k}+3 a_{k +2}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +3}=\frac {k^{2} a_{k}-5 k^{2} a_{k +1}+8 k^{2} a_{k +2}+2 k r a_{k}-10 k r a_{k +1}+16 k r a_{k +2}+r^{2} a_{k}-5 r^{2} a_{k +1}+8 r^{2} a_{k +2}-4 v^{2} a_{k +2}+2 k a_{k}-17 k a_{k +1}+38 k a_{k +2}+2 r a_{k}-17 r a_{k +1}+38 r a_{k +2}-4 v a_{k +2}+a_{k}-15 a_{k +1}+45 a_{k +2}}{4 k^{2}+8 k r +4 r^{2}-4 v^{2}+24 k +24 r -4 v +35} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-v -\frac {1}{2} \\ {} & {} & a_{k +3}=\frac {k^{2} a_{k}-5 k^{2} a_{k +1}+8 k^{2} a_{k +2}+2 k \left (-v -\frac {1}{2}\right ) a_{k}-10 k \left (-v -\frac {1}{2}\right ) a_{k +1}+16 k \left (-v -\frac {1}{2}\right ) a_{k +2}+\left (-v -\frac {1}{2}\right )^{2} a_{k}-5 \left (-v -\frac {1}{2}\right )^{2} a_{k +1}+8 \left (-v -\frac {1}{2}\right )^{2} a_{k +2}-4 v^{2} a_{k +2}+2 k a_{k}-17 k a_{k +1}+38 k a_{k +2}+2 \left (-v -\frac {1}{2}\right ) a_{k}-17 \left (-v -\frac {1}{2}\right ) a_{k +1}+38 \left (-v -\frac {1}{2}\right ) a_{k +2}-4 v a_{k +2}+a_{k}-15 a_{k +1}+45 a_{k +2}}{4 k^{2}+8 k \left (-v -\frac {1}{2}\right )+4 \left (-v -\frac {1}{2}\right )^{2}-4 v^{2}+24 k -28 v +23} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-v -\frac {1}{2} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k -v -\frac {1}{2}}, a_{k +3}=\frac {k^{2} a_{k}-5 k^{2} a_{k +1}+8 k^{2} a_{k +2}+2 k \left (-v -\frac {1}{2}\right ) a_{k}-10 k \left (-v -\frac {1}{2}\right ) a_{k +1}+16 k \left (-v -\frac {1}{2}\right ) a_{k +2}+\left (-v -\frac {1}{2}\right )^{2} a_{k}-5 \left (-v -\frac {1}{2}\right )^{2} a_{k +1}+8 \left (-v -\frac {1}{2}\right )^{2} a_{k +2}-4 v^{2} a_{k +2}+2 k a_{k}-17 k a_{k +1}+38 k a_{k +2}+2 \left (-v -\frac {1}{2}\right ) a_{k}-17 \left (-v -\frac {1}{2}\right ) a_{k +1}+38 \left (-v -\frac {1}{2}\right ) a_{k +2}-4 v a_{k +2}+a_{k}-15 a_{k +1}+45 a_{k +2}}{4 k^{2}+8 k \left (-v -\frac {1}{2}\right )+4 \left (-v -\frac {1}{2}\right )^{2}-4 v^{2}+24 k -28 v +23}, a_{1}=\frac {a_{0} \left (8 \left (-v -\frac {1}{2}\right )^{2}-4 v^{2}-10 v -2\right )}{4 \left (-v -\frac {1}{2}\right )^{2}-4 v^{2}-12 v -1}, a_{2}=\frac {a_{0} \left (44 \left (-v -\frac {1}{2}\right )^{4}-44 \left (-v -\frac {1}{2}\right )^{2} v^{2}+16 v^{4}+156 \left (-v -\frac {1}{2}\right )^{3}-44 \left (-v -\frac {1}{2}\right )^{2} v -84 v^{2} \left (-v -\frac {1}{2}\right )+32 v^{3}+177 \left (-v -\frac {1}{2}\right )^{2}-84 v \left (-v -\frac {1}{2}\right )-36 v^{2}-119 v -\frac {55}{2}\right )}{16 \left (-v -\frac {1}{2}\right )^{4}-32 \left (-v -\frac {1}{2}\right )^{2} v^{2}+16 v^{4}+96 \left (-v -\frac {1}{2}\right )^{3}-32 \left (-v -\frac {1}{2}\right )^{2} v -96 v^{2} \left (-v -\frac {1}{2}\right )+32 v^{3}+200 \left (-v -\frac {1}{2}\right )^{2}-96 v \left (-v -\frac {1}{2}\right )-56 v^{2}-240 v -39}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +1 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +1\right )^{k -v -\frac {1}{2}}, a_{k +3}=\frac {k^{2} a_{k}-5 k^{2} a_{k +1}+8 k^{2} a_{k +2}+2 k \left (-v -\frac {1}{2}\right ) a_{k}-10 k \left (-v -\frac {1}{2}\right ) a_{k +1}+16 k \left (-v -\frac {1}{2}\right ) a_{k +2}+\left (-v -\frac {1}{2}\right )^{2} a_{k}-5 \left (-v -\frac {1}{2}\right )^{2} a_{k +1}+8 \left (-v -\frac {1}{2}\right )^{2} a_{k +2}-4 v^{2} a_{k +2}+2 k a_{k}-17 k a_{k +1}+38 k a_{k +2}+2 \left (-v -\frac {1}{2}\right ) a_{k}-17 \left (-v -\frac {1}{2}\right ) a_{k +1}+38 \left (-v -\frac {1}{2}\right ) a_{k +2}-4 v a_{k +2}+a_{k}-15 a_{k +1}+45 a_{k +2}}{4 k^{2}+8 k \left (-v -\frac {1}{2}\right )+4 \left (-v -\frac {1}{2}\right )^{2}-4 v^{2}+24 k -28 v +23}, a_{1}=\frac {a_{0} \left (8 \left (-v -\frac {1}{2}\right )^{2}-4 v^{2}-10 v -2\right )}{4 \left (-v -\frac {1}{2}\right )^{2}-4 v^{2}-12 v -1}, a_{2}=\frac {a_{0} \left (44 \left (-v -\frac {1}{2}\right )^{4}-44 \left (-v -\frac {1}{2}\right )^{2} v^{2}+16 v^{4}+156 \left (-v -\frac {1}{2}\right )^{3}-44 \left (-v -\frac {1}{2}\right )^{2} v -84 v^{2} \left (-v -\frac {1}{2}\right )+32 v^{3}+177 \left (-v -\frac {1}{2}\right )^{2}-84 v \left (-v -\frac {1}{2}\right )-36 v^{2}-119 v -\frac {55}{2}\right )}{16 \left (-v -\frac {1}{2}\right )^{4}-32 \left (-v -\frac {1}{2}\right )^{2} v^{2}+16 v^{4}+96 \left (-v -\frac {1}{2}\right )^{3}-32 \left (-v -\frac {1}{2}\right )^{2} v -96 v^{2} \left (-v -\frac {1}{2}\right )+32 v^{3}+200 \left (-v -\frac {1}{2}\right )^{2}-96 v \left (-v -\frac {1}{2}\right )-56 v^{2}-240 v -39}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =v +\frac {1}{2} \\ {} & {} & a_{k +3}=\frac {k^{2} a_{k}-5 k^{2} a_{k +1}+8 k^{2} a_{k +2}+2 k \left (v +\frac {1}{2}\right ) a_{k}-10 k \left (v +\frac {1}{2}\right ) a_{k +1}+16 k \left (v +\frac {1}{2}\right ) a_{k +2}+\left (v +\frac {1}{2}\right )^{2} a_{k}-5 \left (v +\frac {1}{2}\right )^{2} a_{k +1}+8 \left (v +\frac {1}{2}\right )^{2} a_{k +2}-4 v^{2} a_{k +2}+2 k a_{k}-17 k a_{k +1}+38 k a_{k +2}+2 \left (v +\frac {1}{2}\right ) a_{k}-17 \left (v +\frac {1}{2}\right ) a_{k +1}+38 \left (v +\frac {1}{2}\right ) a_{k +2}-4 v a_{k +2}+a_{k}-15 a_{k +1}+45 a_{k +2}}{4 k^{2}+8 k \left (v +\frac {1}{2}\right )+4 \left (v +\frac {1}{2}\right )^{2}-4 v^{2}+24 k +20 v +47} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =v +\frac {1}{2} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +v +\frac {1}{2}}, a_{k +3}=\frac {k^{2} a_{k}-5 k^{2} a_{k +1}+8 k^{2} a_{k +2}+2 k \left (v +\frac {1}{2}\right ) a_{k}-10 k \left (v +\frac {1}{2}\right ) a_{k +1}+16 k \left (v +\frac {1}{2}\right ) a_{k +2}+\left (v +\frac {1}{2}\right )^{2} a_{k}-5 \left (v +\frac {1}{2}\right )^{2} a_{k +1}+8 \left (v +\frac {1}{2}\right )^{2} a_{k +2}-4 v^{2} a_{k +2}+2 k a_{k}-17 k a_{k +1}+38 k a_{k +2}+2 \left (v +\frac {1}{2}\right ) a_{k}-17 \left (v +\frac {1}{2}\right ) a_{k +1}+38 \left (v +\frac {1}{2}\right ) a_{k +2}-4 v a_{k +2}+a_{k}-15 a_{k +1}+45 a_{k +2}}{4 k^{2}+8 k \left (v +\frac {1}{2}\right )+4 \left (v +\frac {1}{2}\right )^{2}-4 v^{2}+24 k +20 v +47}, a_{1}=\frac {a_{0} \left (8 \left (v +\frac {1}{2}\right )^{2}-4 v^{2}+2 v +4\right )}{4 \left (v +\frac {1}{2}\right )^{2}-4 v^{2}+4 v +7}, a_{2}=\frac {a_{0} \left (44 \left (v +\frac {1}{2}\right )^{4}-44 \left (v +\frac {1}{2}\right )^{2} v^{2}+16 v^{4}+156 \left (v +\frac {1}{2}\right )^{3}-44 \left (v +\frac {1}{2}\right )^{2} v -84 v^{2} \left (v +\frac {1}{2}\right )+32 v^{3}+177 \left (v +\frac {1}{2}\right )^{2}-84 v \left (v +\frac {1}{2}\right )-36 v^{2}+15 v +\frac {79}{2}\right )}{16 \left (v +\frac {1}{2}\right )^{4}-32 \left (v +\frac {1}{2}\right )^{2} v^{2}+16 v^{4}+96 \left (v +\frac {1}{2}\right )^{3}-32 \left (v +\frac {1}{2}\right )^{2} v -96 v^{2} \left (v +\frac {1}{2}\right )+32 v^{3}+200 \left (v +\frac {1}{2}\right )^{2}-96 v \left (v +\frac {1}{2}\right )-56 v^{2}+96 v +129}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +1 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +1\right )^{k +v +\frac {1}{2}}, a_{k +3}=\frac {k^{2} a_{k}-5 k^{2} a_{k +1}+8 k^{2} a_{k +2}+2 k \left (v +\frac {1}{2}\right ) a_{k}-10 k \left (v +\frac {1}{2}\right ) a_{k +1}+16 k \left (v +\frac {1}{2}\right ) a_{k +2}+\left (v +\frac {1}{2}\right )^{2} a_{k}-5 \left (v +\frac {1}{2}\right )^{2} a_{k +1}+8 \left (v +\frac {1}{2}\right )^{2} a_{k +2}-4 v^{2} a_{k +2}+2 k a_{k}-17 k a_{k +1}+38 k a_{k +2}+2 \left (v +\frac {1}{2}\right ) a_{k}-17 \left (v +\frac {1}{2}\right ) a_{k +1}+38 \left (v +\frac {1}{2}\right ) a_{k +2}-4 v a_{k +2}+a_{k}-15 a_{k +1}+45 a_{k +2}}{4 k^{2}+8 k \left (v +\frac {1}{2}\right )+4 \left (v +\frac {1}{2}\right )^{2}-4 v^{2}+24 k +20 v +47}, a_{1}=\frac {a_{0} \left (8 \left (v +\frac {1}{2}\right )^{2}-4 v^{2}+2 v +4\right )}{4 \left (v +\frac {1}{2}\right )^{2}-4 v^{2}+4 v +7}, a_{2}=\frac {a_{0} \left (44 \left (v +\frac {1}{2}\right )^{4}-44 \left (v +\frac {1}{2}\right )^{2} v^{2}+16 v^{4}+156 \left (v +\frac {1}{2}\right )^{3}-44 \left (v +\frac {1}{2}\right )^{2} v -84 v^{2} \left (v +\frac {1}{2}\right )+32 v^{3}+177 \left (v +\frac {1}{2}\right )^{2}-84 v \left (v +\frac {1}{2}\right )-36 v^{2}+15 v +\frac {79}{2}\right )}{16 \left (v +\frac {1}{2}\right )^{4}-32 \left (v +\frac {1}{2}\right )^{2} v^{2}+16 v^{4}+96 \left (v +\frac {1}{2}\right )^{3}-32 \left (v +\frac {1}{2}\right )^{2} v -96 v^{2} \left (v +\frac {1}{2}\right )+32 v^{3}+200 \left (v +\frac {1}{2}\right )^{2}-96 v \left (v +\frac {1}{2}\right )-56 v^{2}+96 v +129}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +1\right )^{k -v -\frac {1}{2}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} \left (x +1\right )^{k +v +\frac {1}{2}}\right ), a_{k +3}=\frac {k^{2} a_{k}-5 k^{2} a_{k +1}+8 k^{2} a_{k +2}+2 k \left (-v -\frac {1}{2}\right ) a_{k}-10 k \left (-v -\frac {1}{2}\right ) a_{k +1}+16 k \left (-v -\frac {1}{2}\right ) a_{k +2}+\left (-v -\frac {1}{2}\right )^{2} a_{k}-5 \left (-v -\frac {1}{2}\right )^{2} a_{k +1}+8 \left (-v -\frac {1}{2}\right )^{2} a_{k +2}-4 v^{2} a_{k +2}+2 k a_{k}-17 k a_{k +1}+38 k a_{k +2}+2 \left (-v -\frac {1}{2}\right ) a_{k}-17 \left (-v -\frac {1}{2}\right ) a_{k +1}+38 \left (-v -\frac {1}{2}\right ) a_{k +2}-4 v a_{k +2}+a_{k}-15 a_{k +1}+45 a_{k +2}}{4 k^{2}+8 k \left (-v -\frac {1}{2}\right )+4 \left (-v -\frac {1}{2}\right )^{2}-4 v^{2}+24 k -28 v +23}, a_{1}=\frac {a_{0} \left (8 \left (-v -\frac {1}{2}\right )^{2}-4 v^{2}-10 v -2\right )}{4 \left (-v -\frac {1}{2}\right )^{2}-4 v^{2}-12 v -1}, a_{2}=\frac {a_{0} \left (44 \left (-v -\frac {1}{2}\right )^{4}-44 \left (-v -\frac {1}{2}\right )^{2} v^{2}+16 v^{4}+156 \left (-v -\frac {1}{2}\right )^{3}-44 \left (-v -\frac {1}{2}\right )^{2} v -84 v^{2} \left (-v -\frac {1}{2}\right )+32 v^{3}+177 \left (-v -\frac {1}{2}\right )^{2}-84 v \left (-v -\frac {1}{2}\right )-36 v^{2}-119 v -\frac {55}{2}\right )}{16 \left (-v -\frac {1}{2}\right )^{4}-32 \left (-v -\frac {1}{2}\right )^{2} v^{2}+16 v^{4}+96 \left (-v -\frac {1}{2}\right )^{3}-32 \left (-v -\frac {1}{2}\right )^{2} v -96 v^{2} \left (-v -\frac {1}{2}\right )+32 v^{3}+200 \left (-v -\frac {1}{2}\right )^{2}-96 v \left (-v -\frac {1}{2}\right )-56 v^{2}-240 v -39}, b_{k +3}=\frac {k^{2} b_{k}-5 k^{2} b_{k +1}+8 k^{2} b_{k +2}+2 k \left (v +\frac {1}{2}\right ) b_{k}-10 k \left (v +\frac {1}{2}\right ) b_{k +1}+16 k \left (v +\frac {1}{2}\right ) b_{k +2}+\left (v +\frac {1}{2}\right )^{2} b_{k}-5 \left (v +\frac {1}{2}\right )^{2} b_{k +1}+8 \left (v +\frac {1}{2}\right )^{2} b_{k +2}-4 v^{2} b_{k +2}+2 k b_{k}-17 k b_{k +1}+38 k b_{k +2}+2 \left (v +\frac {1}{2}\right ) b_{k}-17 \left (v +\frac {1}{2}\right ) b_{k +1}+38 \left (v +\frac {1}{2}\right ) b_{k +2}-4 v b_{k +2}+b_{k}-15 b_{k +1}+45 b_{k +2}}{4 k^{2}+8 k \left (v +\frac {1}{2}\right )+4 \left (v +\frac {1}{2}\right )^{2}-4 v^{2}+24 k +20 v +47}, b_{1}=\frac {b_{0} \left (8 \left (v +\frac {1}{2}\right )^{2}-4 v^{2}+2 v +4\right )}{4 \left (v +\frac {1}{2}\right )^{2}-4 v^{2}+4 v +7}, b_{2}=\frac {b_{0} \left (44 \left (v +\frac {1}{2}\right )^{4}-44 \left (v +\frac {1}{2}\right )^{2} v^{2}+16 v^{4}+156 \left (v +\frac {1}{2}\right )^{3}-44 \left (v +\frac {1}{2}\right )^{2} v -84 v^{2} \left (v +\frac {1}{2}\right )+32 v^{3}+177 \left (v +\frac {1}{2}\right )^{2}-84 v \left (v +\frac {1}{2}\right )-36 v^{2}+15 v +\frac {79}{2}\right )}{16 \left (v +\frac {1}{2}\right )^{4}-32 \left (v +\frac {1}{2}\right )^{2} v^{2}+16 v^{4}+96 \left (v +\frac {1}{2}\right )^{3}-32 \left (v +\frac {1}{2}\right )^{2} v -96 v^{2} \left (v +\frac {1}{2}\right )+32 v^{3}+200 \left (v +\frac {1}{2}\right )^{2}-96 v \left (v +\frac {1}{2}\right )-56 v^{2}+96 v +129}\right ] \end {array} \]

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying a Liouvillian solution using Kovacics algorithm 
<- No Liouvillian solutions exists 
-> Trying a solution in terms of special functions: 
   -> Bessel 
   -> elliptic 
   -> Legendre 
   -> Kummer 
      -> hyper3: Equivalence to 1F1 under a power @ Moebius 
   -> hypergeometric 
      -> heuristic approach 
      <- heuristic approach successful 
      -> solution has integrals; searching for one without integrals... 
         -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
         <- hyper3 successful: received ODE is equivalent to the 2F1 ODE 
      <- hypergeometric solution without integrals succesful 
   <- hypergeometric successful 
<- special function solution successful`
 

✓ Solution by Maple

Time used: 0.297 (sec). Leaf size: 69

dsolve(diff(diff(y(x),x),x) = -1/(x^2-1)*(3*x^2-1)/x*diff(y(x),x)-(x^2-1-(2*v+1)^2)/(x^2-1)^2*y(x),y(x), singsol=all)
 

\[ y \left (x \right ) = c_{1} \left (x^{2}-1\right )^{-v -\frac {1}{2}} \operatorname {hypergeom}\left (\left [-v , -v \right ], \left [-2 v \right ], -x^{2}+1\right )+c_{2} \left (x^{2}-1\right )^{v +\frac {1}{2}} \operatorname {hypergeom}\left (\left [v +1, v +1\right ], \left [2 v +2\right ], -x^{2}+1\right ) \]

✓ Solution by Mathematica

Time used: 0.22 (sec). Leaf size: 72

DSolve[y''[x] == -(((-1 - (1 + 2*v)^2 + x^2)*y[x])/(-1 + x^2)^2) - ((-1 + 3*x^2)*y'[x])/(x*(-1 + x^2)),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \left (x^2-1\right )^{-v-\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (-v,-v,-2 v,1-x^2\right )+c_2 \left (x^2-1\right )^{v+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (v+1,v+1,2 v+2,1-x^2\right ) \]