Internal problem ID [9702]
Internal file name [OUTPUT/8644_Monday_June_06_2022_04_36_03_AM_21799285/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1375.
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.
\[ \boxed {y^{\prime \prime }+\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}+\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}}=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) \left (x^{4}-2 x^{2}+1\right )-2 x \left (-n -1+2 a \right ) \left (x +1\right ) \left (x -1\right ) y^{\prime }+\left (\left (n^{2}+\left (-4 a +1\right ) n +4 a^{2}-v^{2}-2 a -v \right ) x^{2}-n^{2}+v^{2}+2 a -n +v \right ) y=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 a^{2} x^{2}-4 a \,x^{2} n +n^{2} x^{2}-v^{2} x^{2}-2 a \,x^{2}+n \,x^{2}-v \,x^{2}-n^{2}+v^{2}+2 a -n +v \right ) y}{x^{4}-2 x^{2}+1}+\frac {2 y^{\prime } x \left (-n -1+2 a \right )}{x^{2}-1} \\ \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 {2 y^{\prime } x \left (-n -1+2 a \right )}{x^{2}-1}+\frac {\left (4 a^{2} x^{2}-4 a \,x^{2} n +n^{2} x^{2}-v^{2} x^{2}-2 a \,x^{2}+n \,x^{2}-v \,x^{2}-n^{2}+v^{2}+2 a -n +v \right ) y}{x^{4}-2 x^{2}+1}=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 {2 x \left (-n -1+2 a \right )}{x^{2}-1}, P_{3}\left (x \right )=\frac {4 a^{2} x^{2}-4 a \,x^{2} n +n^{2} x^{2}-v^{2} x^{2}-2 a \,x^{2}+n \,x^{2}-v \,x^{2}-n^{2}+v^{2}+2 a -n +v}{x^{4}-2 x^{2}+1}\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}}}=n +1-2 a \\ {} & \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}}}=a^{2}-n a \\ {} & \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^{2}-1\right ) \left (x^{4}-2 x^{2}+1\right )-2 x \left (-n -1+2 a \right ) \left (x^{4}-2 x^{2}+1\right ) y^{\prime }+\left (4 a^{2} x^{2}-4 a \,x^{2} n +n^{2} x^{2}-v^{2} x^{2}-2 a \,x^{2}+n \,x^{2}-v \,x^{2}-n^{2}+v^{2}+2 a -n +v \right ) \left (x^{2}-1\right ) y=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^{6}-6 u^{5}+12 u^{4}-8 u^{3}\right ) \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )+\left (-4 a \,u^{5}+2 n \,u^{5}+20 a \,u^{4}-10 n \,u^{4}+2 u^{5}-32 a \,u^{3}+16 n \,u^{3}-10 u^{4}+16 a \,u^{2}-8 n \,u^{2}+16 u^{3}-8 u^{2}\right ) \left (\frac {d}{d u}y \left (u \right )\right )+\left (4 a^{2} u^{4}-4 a n \,u^{4}+n^{2} u^{4}-u^{4} v^{2}-16 a^{2} u^{3}+16 a n \,u^{3}-2 a \,u^{4}-4 n^{2} u^{3}+n \,u^{4}-u^{4} v +4 u^{3} v^{2}+20 a^{2} u^{2}-20 a n \,u^{2}+8 a \,u^{3}+4 n^{2} u^{2}-4 n \,u^{3}+4 u^{3} v -4 u^{2} v^{2}-8 a^{2} u +8 a n u -8 a \,u^{2}+4 n \,u^{2}-4 u^{2} v \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 =1..4 \\ {} & {} & 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 =2..5 \\ {} & {} & 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 =3..6 \\ {} & {} & 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} \\ {} & {} & -8 a_{0} \left (a -r \right ) \left (a -n -r \right ) u^{1+r}+\left (-8 a_{1} \left (a -1-r \right ) \left (a -1-n -r \right )+4 a_{0} \left (5 a^{2}-5 n a -8 a r +n^{2}+4 n r +3 r^{2}-v^{2}-2 a +n +r -v \right )\right ) u^{2+r}+\left (-8 a_{2} \left (a -2-r \right ) \left (a -2-n -r \right )+4 a_{1} \left (5 a^{2}-5 n a -8 a r +n^{2}+4 n r +3 r^{2}-v^{2}-10 a +5 n +7 r -v +4\right )-2 a_{0} \left (8 a^{2}-8 n a -10 a r +2 n^{2}+5 n r +3 r^{2}-2 v^{2}-4 a +2 n +2 r -2 v \right )\right ) u^{3+r}+\left (\moverset {\infty }{\munderset {k =4}{\sum }}\left (-8 a_{k -1} \left (a -k +1-r \right ) \left (a -k +1-n -r \right )+4 a_{k -2} \left (5 a^{2}-8 a \left (k -2\right )-5 n a -8 a r +3 \left (k -2\right )^{2}+4 n \left (k -2\right )+6 \left (k -2\right ) r +n^{2}+4 n r +3 r^{2}-v^{2}-2 a +k -2+n +r -v \right )-2 a_{k -3} \left (8 a^{2}-10 a \left (k -3\right )-8 n a -10 a r +3 \left (k -3\right )^{2}+5 n \left (k -3\right )+6 \left (k -3\right ) r +2 n^{2}+5 n r +3 r^{2}-2 v^{2}-4 a +2 k -6+2 n +2 r -2 v \right )+a_{k -4} \left (2 a -k +4-n -r +v \right ) \left (-v +3-r -n -k +2 a \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -8 \left (a -r \right ) \left (a -n -r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{a , a -n \right \} \\ \bullet & {} & \textrm {The coefficients of each power of}\hspace {3pt} u \hspace {3pt}\textrm {must be 0}\hspace {3pt} \\ {} & {} & \left [-8 a_{1} \left (a -1-r \right ) \left (a -1-n -r \right )+4 a_{0} \left (5 a^{2}-5 n a -8 a r +n^{2}+4 n r +3 r^{2}-v^{2}-2 a +n +r -v \right )=0, -8 a_{2} \left (a -2-r \right ) \left (a -2-n -r \right )+4 a_{1} \left (5 a^{2}-5 n a -8 a r +n^{2}+4 n r +3 r^{2}-v^{2}-10 a +5 n +7 r -v +4\right )-2 a_{0} \left (8 a^{2}-8 n a -10 a r +2 n^{2}+5 n r +3 r^{2}-2 v^{2}-4 a +2 n +2 r -2 v \right )=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{1}=\frac {a_{0} \left (5 a^{2}-5 n a -8 a r +n^{2}+4 n r +3 r^{2}-v^{2}-2 a +n +r -v \right )}{2 \left (a^{2}-n a -2 a r +n r +r^{2}-2 a +n +2 r +1\right )}, a_{2}=\frac {a_{0} \left (17 a^{4}-34 n \,a^{3}-54 a^{3} r +25 n^{2} a^{2}+81 a^{2} n r +63 a^{2} r^{2}-8 v^{2} a^{2}-8 n^{3} a -39 a \,n^{2} r -63 a n \,r^{2}+8 v^{2} n a -32 a \,r^{3}+12 a r \,v^{2}+n^{4}+6 n^{3} r +15 n^{2} r^{2}-2 n^{2} v^{2}+16 n \,r^{3}-6 n r \,v^{2}+6 r^{4}-4 r^{2} v^{2}+v^{4}-40 a^{3}+60 n \,a^{2}+90 a^{2} r -8 v \,a^{2}-28 a \,n^{2}-90 a n r +8 v n a -66 a \,r^{2}+12 a r v +8 a \,v^{2}+4 n^{3}+21 n^{2} r -2 n^{2} v +33 n \,r^{2}-6 n r v -4 n \,v^{2}+16 r^{3}-4 r^{2} v -4 r \,v^{2}+2 v^{3}+24 a^{2}-24 n a -34 a r +8 a v +5 n^{2}+17 n r -4 v n +12 r^{2}-4 r v -v^{2}-4 a +2 n +2 r -2 v \right )}{4 \left (a^{4}-2 n \,a^{3}-4 a^{3} r +n^{2} a^{2}+6 a^{2} n r +6 a^{2} r^{2}-2 a \,n^{2} r -6 a n \,r^{2}-4 a \,r^{3}+n^{2} r^{2}+2 n \,r^{3}+r^{4}-6 a^{3}+9 n \,a^{2}+18 a^{2} r -3 a \,n^{2}-18 a n r -18 a \,r^{2}+3 n^{2} r +9 n \,r^{2}+6 r^{3}+13 a^{2}-13 n a -26 a r +2 n^{2}+13 n r +13 r^{2}-12 a +6 n +12 r +4\right )}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (4 a_{k -4}-16 a_{k -3}+20 a_{k -2}-8 a_{k -1}\right ) a^{2}+\left (\left (-4 a_{k -4}+20 a_{k -3}-32 a_{k -2}+16 a_{k -1}\right ) k +\left (-4 a_{k -4}+20 a_{k -3}-32 a_{k -2}+16 a_{k -1}\right ) r +\left (-4 a_{k -4}+16 a_{k -3}-20 a_{k -2}+8 a_{k -1}\right ) n +14 a_{k -4}-52 a_{k -3}+56 a_{k -2}-16 a_{k -1}\right ) a +\left (a_{k -4}-6 a_{k -3}+12 a_{k -2}-8 a_{k -1}\right ) k^{2}+\left (\left (2 a_{k -4}-12 a_{k -3}+24 a_{k -2}-16 a_{k -1}\right ) r +\left (2 a_{k -4}-10 a_{k -3}+16 a_{k -2}-8 a_{k -1}\right ) n -7 a_{k -4}+32 a_{k -3}-44 a_{k -2}+16 a_{k -1}\right ) k +\left (a_{k -4}-6 a_{k -3}+12 a_{k -2}-8 a_{k -1}\right ) r^{2}+\left (\left (2 a_{k -4}-10 a_{k -3}+16 a_{k -2}-8 a_{k -1}\right ) n -7 a_{k -4}+32 a_{k -3}-44 a_{k -2}+16 a_{k -1}\right ) r +\left (a_{k -4}-4 a_{k -3}+4 a_{k -2}\right ) n^{2}+\left (-7 a_{k -4}+26 a_{k -3}-28 a_{k -2}+8 a_{k -1}\right ) n +\left (-v^{2}-v +12\right ) a_{k -4}+\left (4 v^{2}+4 v -42\right ) a_{k -3}+\left (-4 v^{2}-4 v +40\right ) a_{k -2}-8 a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +4 \\ {} & {} & \left (4 a_{k}-16 a_{k +1}+20 a_{k +2}-8 a_{k +3}\right ) a^{2}+\left (\left (-4 a_{k}+20 a_{k +1}-32 a_{k +2}+16 a_{k +3}\right ) \left (k +4\right )+\left (-4 a_{k}+20 a_{k +1}-32 a_{k +2}+16 a_{k +3}\right ) r +\left (-4 a_{k}+16 a_{k +1}-20 a_{k +2}+8 a_{k +3}\right ) n +14 a_{k}-52 a_{k +1}+56 a_{k +2}-16 a_{k +3}\right ) a +\left (a_{k}-6 a_{k +1}+12 a_{k +2}-8 a_{k +3}\right ) \left (k +4\right )^{2}+\left (\left (2 a_{k}-12 a_{k +1}+24 a_{k +2}-16 a_{k +3}\right ) r +\left (2 a_{k}-10 a_{k +1}+16 a_{k +2}-8 a_{k +3}\right ) n -7 a_{k}+32 a_{k +1}-44 a_{k +2}+16 a_{k +3}\right ) \left (k +4\right )+\left (a_{k}-6 a_{k +1}+12 a_{k +2}-8 a_{k +3}\right ) r^{2}+\left (\left (2 a_{k}-10 a_{k +1}+16 a_{k +2}-8 a_{k +3}\right ) n -7 a_{k}+32 a_{k +1}-44 a_{k +2}+16 a_{k +3}\right ) r +\left (a_{k}-4 a_{k +1}+4 a_{k +2}\right ) n^{2}+\left (-7 a_{k}+26 a_{k +1}-28 a_{k +2}+8 a_{k +3}\right ) n +\left (-v^{2}-v +12\right ) a_{k}+\left (4 v^{2}+4 v -42\right ) a_{k +1}+\left (-4 v^{2}-4 v +40\right ) a_{k +2}-8 a_{k +3}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +3}=\frac {4 a_{k} a^{2}-16 a^{2} a_{k +1}+20 a^{2} a_{k +2}-4 a k a_{k}+20 a k a_{k +1}-32 a k a_{k +2}-4 a_{k} a n +16 a n a_{k +1}-20 a n a_{k +2}-4 a r a_{k}+20 a r a_{k +1}-32 a r a_{k +2}+k^{2} a_{k}-6 k^{2} a_{k +1}+12 k^{2} a_{k +2}+2 k n a_{k}-10 k n a_{k +1}+16 k n a_{k +2}+2 k r a_{k}-12 k r a_{k +1}+24 k r a_{k +2}+a_{k} n^{2}-4 n^{2} a_{k +1}+4 n^{2} a_{k +2}+2 n r a_{k}-10 n r a_{k +1}+16 n r a_{k +2}+r^{2} a_{k}-6 r^{2} a_{k +1}+12 r^{2} a_{k +2}-a_{k} v^{2}+4 v^{2} a_{k +1}-4 v^{2} a_{k +2}-2 a_{k} a +28 a a_{k +1}-72 a a_{k +2}+k a_{k}-16 k a_{k +1}+52 k a_{k +2}+a_{k} n -14 n a_{k +1}+36 n a_{k +2}+r a_{k}-16 r a_{k +1}+52 r a_{k +2}-a_{k} v +4 v a_{k +1}-4 v a_{k +2}-10 a_{k +1}+56 a_{k +2}}{8 \left (a^{2}-2 a k -n a -2 a r +k^{2}+n k +2 k r +n r +r^{2}-6 a +6 k +3 n +6 r +9\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =a \\ {} & {} & a_{k +3}=\frac {a_{k} a^{2}-2 a^{2} a_{k +1}-2 a k a_{k}+8 a k a_{k +1}-8 a k a_{k +2}-2 a_{k} a n +6 a n a_{k +1}-4 a n a_{k +2}+k^{2} a_{k}-6 k^{2} a_{k +1}+12 k^{2} a_{k +2}+2 k n a_{k}-10 k n a_{k +1}+16 k n a_{k +2}+a_{k} n^{2}-4 n^{2} a_{k +1}+4 n^{2} a_{k +2}-a_{k} v^{2}+4 v^{2} a_{k +1}-4 v^{2} a_{k +2}-a_{k} a +12 a a_{k +1}-20 a a_{k +2}+k a_{k}-16 k a_{k +1}+52 k a_{k +2}+a_{k} n -14 n a_{k +1}+36 n a_{k +2}-a_{k} v +4 v a_{k +1}-4 v a_{k +2}-10 a_{k +1}+56 a_{k +2}}{8 \left (k^{2}+n k +6 k +3 n +9\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =a \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +a}, a_{k +3}=\frac {a_{k} a^{2}-2 a^{2} a_{k +1}-2 a k a_{k}+8 a k a_{k +1}-8 a k a_{k +2}-2 a_{k} a n +6 a n a_{k +1}-4 a n a_{k +2}+k^{2} a_{k}-6 k^{2} a_{k +1}+12 k^{2} a_{k +2}+2 k n a_{k}-10 k n a_{k +1}+16 k n a_{k +2}+a_{k} n^{2}-4 n^{2} a_{k +1}+4 n^{2} a_{k +2}-a_{k} v^{2}+4 v^{2} a_{k +1}-4 v^{2} a_{k +2}-a_{k} a +12 a a_{k +1}-20 a a_{k +2}+k a_{k}-16 k a_{k +1}+52 k a_{k +2}+a_{k} n -14 n a_{k +1}+36 n a_{k +2}-a_{k} v +4 v a_{k +1}-4 v a_{k +2}-10 a_{k +1}+56 a_{k +2}}{8 \left (k^{2}+n k +6 k +3 n +9\right )}, a_{1}=\frac {a_{0} \left (-n a +n^{2}-v^{2}-a +n -v \right )}{2 \left (n +1\right )}, a_{2}=\frac {a_{0} \left (n^{2} a^{2}-2 n^{3} a +2 v^{2} n a +n^{4}-2 n^{2} v^{2}+v^{4}+3 n \,a^{2}-7 a \,n^{2}+2 v n a +4 a \,v^{2}+4 n^{3}-2 n^{2} v -4 n \,v^{2}+2 v^{3}+2 a^{2}-7 n a +4 a v +5 n^{2}-4 v n -v^{2}-2 a +2 n -2 v \right )}{4 \left (2 n^{2}+6 n +4\right )}\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 +a}, a_{k +3}=\frac {a_{k} a^{2}-2 a^{2} a_{k +1}-2 a k a_{k}+8 a k a_{k +1}-8 a k a_{k +2}-2 a_{k} a n +6 a n a_{k +1}-4 a n a_{k +2}+k^{2} a_{k}-6 k^{2} a_{k +1}+12 k^{2} a_{k +2}+2 k n a_{k}-10 k n a_{k +1}+16 k n a_{k +2}+a_{k} n^{2}-4 n^{2} a_{k +1}+4 n^{2} a_{k +2}-a_{k} v^{2}+4 v^{2} a_{k +1}-4 v^{2} a_{k +2}-a_{k} a +12 a a_{k +1}-20 a a_{k +2}+k a_{k}-16 k a_{k +1}+52 k a_{k +2}+a_{k} n -14 n a_{k +1}+36 n a_{k +2}-a_{k} v +4 v a_{k +1}-4 v a_{k +2}-10 a_{k +1}+56 a_{k +2}}{8 \left (k^{2}+n k +6 k +3 n +9\right )}, a_{1}=\frac {a_{0} \left (-n a +n^{2}-v^{2}-a +n -v \right )}{2 \left (n +1\right )}, a_{2}=\frac {a_{0} \left (n^{2} a^{2}-2 n^{3} a +2 v^{2} n a +n^{4}-2 n^{2} v^{2}+v^{4}+3 n \,a^{2}-7 a \,n^{2}+2 v n a +4 a \,v^{2}+4 n^{3}-2 n^{2} v -4 n \,v^{2}+2 v^{3}+2 a^{2}-7 n a +4 a v +5 n^{2}-4 v n -v^{2}-2 a +2 n -2 v \right )}{4 \left (2 n^{2}+6 n +4\right )}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =a -n \\ {} & {} & a_{k +3}=\frac {-4 a \left (a -n \right ) a_{k}+20 a \left (a -n \right ) a_{k +1}-32 a \left (a -n \right ) a_{k +2}+2 k \left (a -n \right ) a_{k}-12 k \left (a -n \right ) a_{k +1}+24 k \left (a -n \right ) a_{k +2}+2 n \left (a -n \right ) a_{k}-10 n \left (a -n \right ) a_{k +1}+16 n \left (a -n \right ) a_{k +2}-4 a_{k} a n +a_{k} n^{2}-a_{k} v^{2}+4 a_{k} a^{2}-2 a_{k} a +a_{k} n -a_{k} v +2 k n a_{k}-10 k n a_{k +1}+16 k n a_{k +2}-16 a^{2} a_{k +1}+20 a^{2} a_{k +2}+k^{2} a_{k}-6 k^{2} a_{k +1}+12 k^{2} a_{k +2}-4 n^{2} a_{k +1}+4 n^{2} a_{k +2}+4 v^{2} a_{k +1}-4 v^{2} a_{k +2}+28 a a_{k +1}-72 a a_{k +2}+k a_{k}-16 k a_{k +1}+52 k a_{k +2}-14 n a_{k +1}+36 n a_{k +2}+4 v a_{k +1}-4 v a_{k +2}+\left (a -n \right )^{2} a_{k}-6 \left (a -n \right )^{2} a_{k +1}+12 \left (a -n \right )^{2} a_{k +2}+\left (a -n \right ) a_{k}-16 \left (a -n \right ) a_{k +1}+52 \left (a -n \right ) a_{k +2}-4 a k a_{k}+20 a k a_{k +1}-32 a k a_{k +2}+16 a n a_{k +1}-20 a n a_{k +2}-10 a_{k +1}+56 a_{k +2}}{8 \left (a^{2}-2 a k -n a -2 a \left (a -n \right )+k^{2}+n k +2 k \left (a -n \right )+n \left (a -n \right )+\left (a -n \right )^{2}+6 k -3 n +9\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =a -n \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +a -n}, a_{k +3}=\frac {-4 a \left (a -n \right ) a_{k}+20 a \left (a -n \right ) a_{k +1}-32 a \left (a -n \right ) a_{k +2}+2 k \left (a -n \right ) a_{k}-12 k \left (a -n \right ) a_{k +1}+24 k \left (a -n \right ) a_{k +2}+2 n \left (a -n \right ) a_{k}-10 n \left (a -n \right ) a_{k +1}+16 n \left (a -n \right ) a_{k +2}-4 a_{k} a n +a_{k} n^{2}-a_{k} v^{2}+4 a_{k} a^{2}-2 a_{k} a +a_{k} n -a_{k} v +2 k n a_{k}-10 k n a_{k +1}+16 k n a_{k +2}-16 a^{2} a_{k +1}+20 a^{2} a_{k +2}+k^{2} a_{k}-6 k^{2} a_{k +1}+12 k^{2} a_{k +2}-4 n^{2} a_{k +1}+4 n^{2} a_{k +2}+4 v^{2} a_{k +1}-4 v^{2} a_{k +2}+28 a a_{k +1}-72 a a_{k +2}+k a_{k}-16 k a_{k +1}+52 k a_{k +2}-14 n a_{k +1}+36 n a_{k +2}+4 v a_{k +1}-4 v a_{k +2}+\left (a -n \right )^{2} a_{k}-6 \left (a -n \right )^{2} a_{k +1}+12 \left (a -n \right )^{2} a_{k +2}+\left (a -n \right ) a_{k}-16 \left (a -n \right ) a_{k +1}+52 \left (a -n \right ) a_{k +2}-4 a k a_{k}+20 a k a_{k +1}-32 a k a_{k +2}+16 a n a_{k +1}-20 a n a_{k +2}-10 a_{k +1}+56 a_{k +2}}{8 \left (a^{2}-2 a k -n a -2 a \left (a -n \right )+k^{2}+n k +2 k \left (a -n \right )+n \left (a -n \right )+\left (a -n \right )^{2}+6 k -3 n +9\right )}, a_{1}=\frac {a_{0} \left (5 a^{2}-5 n a -8 a \left (a -n \right )+n^{2}+4 n \left (a -n \right )+3 \left (a -n \right )^{2}-v^{2}-a -v \right )}{2 \left (a^{2}-n a -2 a \left (a -n \right )+n \left (a -n \right )+\left (a -n \right )^{2}-n +1\right )}, a_{2}=\frac {a_{0} \left (-2 a -2 v +5 n^{2}-4 v n -v^{2}-28 a \,n^{2}-4 n \,v^{2}+24 a^{2}+6 \left (a -n \right )^{4}+16 \left (a -n \right )^{3}+81 a^{2} n \left (a -n \right )-39 a \,n^{2} \left (a -n \right )-63 a n \left (a -n \right )^{2}+12 a \left (a -n \right ) v^{2}-6 n \left (a -n \right ) v^{2}-90 a n \left (a -n \right )+12 a \left (a -n \right ) v -6 n \left (a -n \right ) v +17 a^{4}+n^{4}+4 n^{3}-54 a^{3} \left (a -n \right )+63 a^{2} \left (a -n \right )^{2}-32 a \left (a -n \right )^{3}+6 n^{3} \left (a -n \right )+15 n^{2} \left (a -n \right )^{2}+16 n \left (a -n \right )^{3}-4 \left (a -n \right )^{2} v^{2}+90 a^{2} \left (a -n \right )-66 a \left (a -n \right )^{2}+21 n^{2} \left (a -n \right )+33 n \left (a -n \right )^{2}-4 \left (a -n \right )^{2} v -4 \left (a -n \right ) v^{2}-4 \left (a -n \right ) v +8 a \,v^{2}+8 a v +v^{4}+2 v^{3}-24 n a -34 a \left (a -n \right )+17 n \left (a -n \right )-2 n^{2} v^{2}-2 n^{2} v -8 v^{2} a^{2}-8 n^{3} a +60 n \,a^{2}-34 n \,a^{3}+25 n^{2} a^{2}+8 v^{2} n a -8 v \,a^{2}+8 v n a +12 \left (a -n \right )^{2}-40 a^{3}\right )}{4 \left (4-6 n +2 n^{2}-3 a \,n^{2}+13 a^{2}+\left (a -n \right )^{4}+6 \left (a -n \right )^{3}+6 a^{2} n \left (a -n \right )-2 a \,n^{2} \left (a -n \right )-6 a n \left (a -n \right )^{2}-18 a n \left (a -n \right )+a^{4}-4 a^{3} \left (a -n \right )+6 a^{2} \left (a -n \right )^{2}-4 a \left (a -n \right )^{3}+n^{2} \left (a -n \right )^{2}+2 n \left (a -n \right )^{3}+18 a^{2} \left (a -n \right )-18 a \left (a -n \right )^{2}+3 n^{2} \left (a -n \right )+9 n \left (a -n \right )^{2}-13 n a -26 a \left (a -n \right )+13 n \left (a -n \right )+9 n \,a^{2}-2 n \,a^{3}+n^{2} a^{2}+13 \left (a -n \right )^{2}-6 a^{3}\right )}\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 +a -n}, a_{k +3}=\frac {-4 a \left (a -n \right ) a_{k}+20 a \left (a -n \right ) a_{k +1}-32 a \left (a -n \right ) a_{k +2}+2 k \left (a -n \right ) a_{k}-12 k \left (a -n \right ) a_{k +1}+24 k \left (a -n \right ) a_{k +2}+2 n \left (a -n \right ) a_{k}-10 n \left (a -n \right ) a_{k +1}+16 n \left (a -n \right ) a_{k +2}-4 a_{k} a n +a_{k} n^{2}-a_{k} v^{2}+4 a_{k} a^{2}-2 a_{k} a +a_{k} n -a_{k} v +2 k n a_{k}-10 k n a_{k +1}+16 k n a_{k +2}-16 a^{2} a_{k +1}+20 a^{2} a_{k +2}+k^{2} a_{k}-6 k^{2} a_{k +1}+12 k^{2} a_{k +2}-4 n^{2} a_{k +1}+4 n^{2} a_{k +2}+4 v^{2} a_{k +1}-4 v^{2} a_{k +2}+28 a a_{k +1}-72 a a_{k +2}+k a_{k}-16 k a_{k +1}+52 k a_{k +2}-14 n a_{k +1}+36 n a_{k +2}+4 v a_{k +1}-4 v a_{k +2}+\left (a -n \right )^{2} a_{k}-6 \left (a -n \right )^{2} a_{k +1}+12 \left (a -n \right )^{2} a_{k +2}+\left (a -n \right ) a_{k}-16 \left (a -n \right ) a_{k +1}+52 \left (a -n \right ) a_{k +2}-4 a k a_{k}+20 a k a_{k +1}-32 a k a_{k +2}+16 a n a_{k +1}-20 a n a_{k +2}-10 a_{k +1}+56 a_{k +2}}{8 \left (a^{2}-2 a k -n a -2 a \left (a -n \right )+k^{2}+n k +2 k \left (a -n \right )+n \left (a -n \right )+\left (a -n \right )^{2}+6 k -3 n +9\right )}, a_{1}=\frac {a_{0} \left (5 a^{2}-5 n a -8 a \left (a -n \right )+n^{2}+4 n \left (a -n \right )+3 \left (a -n \right )^{2}-v^{2}-a -v \right )}{2 \left (a^{2}-n a -2 a \left (a -n \right )+n \left (a -n \right )+\left (a -n \right )^{2}-n +1\right )}, a_{2}=\frac {a_{0} \left (-2 a -2 v +5 n^{2}-4 v n -v^{2}-28 a \,n^{2}-4 n \,v^{2}+24 a^{2}+6 \left (a -n \right )^{4}+16 \left (a -n \right )^{3}+81 a^{2} n \left (a -n \right )-39 a \,n^{2} \left (a -n \right )-63 a n \left (a -n \right )^{2}+12 a \left (a -n \right ) v^{2}-6 n \left (a -n \right ) v^{2}-90 a n \left (a -n \right )+12 a \left (a -n \right ) v -6 n \left (a -n \right ) v +17 a^{4}+n^{4}+4 n^{3}-54 a^{3} \left (a -n \right )+63 a^{2} \left (a -n \right )^{2}-32 a \left (a -n \right )^{3}+6 n^{3} \left (a -n \right )+15 n^{2} \left (a -n \right )^{2}+16 n \left (a -n \right )^{3}-4 \left (a -n \right )^{2} v^{2}+90 a^{2} \left (a -n \right )-66 a \left (a -n \right )^{2}+21 n^{2} \left (a -n \right )+33 n \left (a -n \right )^{2}-4 \left (a -n \right )^{2} v -4 \left (a -n \right ) v^{2}-4 \left (a -n \right ) v +8 a \,v^{2}+8 a v +v^{4}+2 v^{3}-24 n a -34 a \left (a -n \right )+17 n \left (a -n \right )-2 n^{2} v^{2}-2 n^{2} v -8 v^{2} a^{2}-8 n^{3} a +60 n \,a^{2}-34 n \,a^{3}+25 n^{2} a^{2}+8 v^{2} n a -8 v \,a^{2}+8 v n a +12 \left (a -n \right )^{2}-40 a^{3}\right )}{4 \left (4-6 n +2 n^{2}-3 a \,n^{2}+13 a^{2}+\left (a -n \right )^{4}+6 \left (a -n \right )^{3}+6 a^{2} n \left (a -n \right )-2 a \,n^{2} \left (a -n \right )-6 a n \left (a -n \right )^{2}-18 a n \left (a -n \right )+a^{4}-4 a^{3} \left (a -n \right )+6 a^{2} \left (a -n \right )^{2}-4 a \left (a -n \right )^{3}+n^{2} \left (a -n \right )^{2}+2 n \left (a -n \right )^{3}+18 a^{2} \left (a -n \right )-18 a \left (a -n \right )^{2}+3 n^{2} \left (a -n \right )+9 n \left (a -n \right )^{2}-13 n a -26 a \left (a -n \right )+13 n \left (a -n \right )+9 n \,a^{2}-2 n \,a^{3}+n^{2} a^{2}+13 \left (a -n \right )^{2}-6 a^{3}\right )}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} \left (x +1\right )^{k +a}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}c_{k} \left (x +1\right )^{k +a -n}\right ), b_{k +3}=\frac {b_{k} a^{2}-2 a^{2} b_{k +1}-2 a k b_{k}+8 a k b_{k +1}-8 a k b_{k +2}-2 a n b_{k}+6 a n b_{k +1}-4 a n b_{k +2}+k^{2} b_{k}-6 k^{2} b_{k +1}+12 k^{2} b_{k +2}+2 k n b_{k}-10 k n b_{k +1}+16 k n b_{k +2}+n^{2} b_{k}-4 n^{2} b_{k +1}+4 n^{2} b_{k +2}-v^{2} b_{k}+4 v^{2} b_{k +1}-4 v^{2} b_{k +2}-a b_{k}+12 a b_{k +1}-20 a b_{k +2}+k b_{k}-16 k b_{k +1}+52 k b_{k +2}+n b_{k}-14 n b_{k +1}+36 n b_{k +2}-v b_{k}+4 v b_{k +1}-4 v b_{k +2}-10 b_{k +1}+56 b_{k +2}}{8 \left (k^{2}+n k +6 k +3 n +9\right )}, b_{1}=\frac {b_{0} \left (-n a +n^{2}-v^{2}-a +n -v \right )}{2 \left (n +1\right )}, b_{2}=\frac {b_{0} \left (n^{2} a^{2}-2 n^{3} a +2 v^{2} n a +n^{4}-2 n^{2} v^{2}+v^{4}+3 n \,a^{2}-7 a \,n^{2}+2 v n a +4 a \,v^{2}+4 n^{3}-2 n^{2} v -4 n \,v^{2}+2 v^{3}+2 a^{2}-7 n a +4 a v +5 n^{2}-4 v n -v^{2}-2 a +2 n -2 v \right )}{4 \left (2 n^{2}+6 n +4\right )}, c_{k +3}=\frac {4 n^{2} c_{k +2}+36 n c_{k +2}-4 a \left (a -n \right ) c_{k}+20 a \left (a -n \right ) c_{k +1}-32 a \left (a -n \right ) c_{k +2}+2 k \left (a -n \right ) c_{k}-12 k \left (a -n \right ) c_{k +1}+24 k \left (a -n \right ) c_{k +2}+2 n \left (a -n \right ) c_{k}-10 n \left (a -n \right ) c_{k +1}+16 n \left (a -n \right ) c_{k +2}-4 c_{k} a n +2 k n c_{k}-10 k n c_{k +1}+16 k n c_{k +2}+16 a n c_{k +1}-20 a n c_{k +2}+c_{k} n^{2}-c_{k} v^{2}+c_{k} n -c_{k} v -4 n^{2} c_{k +1}+4 v^{2} c_{k +1}-4 v^{2} c_{k +2}-14 n c_{k +1}+4 v c_{k +1}-4 v c_{k +2}+\left (a -n \right )^{2} c_{k}-6 \left (a -n \right )^{2} c_{k +1}+12 \left (a -n \right )^{2} c_{k +2}+\left (a -n \right ) c_{k}-16 \left (a -n \right ) c_{k +1}+52 \left (a -n \right ) c_{k +2}+k^{2} c_{k}-6 k^{2} c_{k +1}+k c_{k}-16 k c_{k +1}+56 c_{k +2}-10 c_{k +1}-4 a k c_{k}+20 a k c_{k +1}-32 a k c_{k +2}+4 c_{k} a^{2}-16 a^{2} c_{k +1}+20 a^{2} c_{k +2}+12 k^{2} c_{k +2}-2 a c_{k}+28 a c_{k +1}-72 a c_{k +2}+52 k c_{k +2}}{8 \left (a^{2}-2 a k -n a -2 a \left (a -n \right )+k^{2}+n k +2 k \left (a -n \right )+n \left (a -n \right )+\left (a -n \right )^{2}+6 k -3 n +9\right )}, c_{1}=\frac {c_{0} \left (5 a^{2}-5 n a -8 a \left (a -n \right )+n^{2}+4 n \left (a -n \right )+3 \left (a -n \right )^{2}-v^{2}-a -v \right )}{2 \left (a^{2}-n a -2 a \left (a -n \right )+n \left (a -n \right )+\left (a -n \right )^{2}-n +1\right )}, c_{2}=\frac {c_{0} \left (-2 a -2 v +5 n^{2}-4 v n -v^{2}-28 a \,n^{2}-4 n \,v^{2}+24 a^{2}+6 \left (a -n \right )^{4}+16 \left (a -n \right )^{3}+81 a^{2} n \left (a -n \right )-39 a \,n^{2} \left (a -n \right )-63 a n \left (a -n \right )^{2}+12 a \left (a -n \right ) v^{2}-6 n \left (a -n \right ) v^{2}-90 a n \left (a -n \right )+12 a \left (a -n \right ) v -6 n \left (a -n \right ) v +17 a^{4}+n^{4}+4 n^{3}-54 a^{3} \left (a -n \right )+63 a^{2} \left (a -n \right )^{2}-32 a \left (a -n \right )^{3}+6 n^{3} \left (a -n \right )+15 n^{2} \left (a -n \right )^{2}+16 n \left (a -n \right )^{3}-4 \left (a -n \right )^{2} v^{2}+90 a^{2} \left (a -n \right )-66 a \left (a -n \right )^{2}+21 n^{2} \left (a -n \right )+33 n \left (a -n \right )^{2}-4 \left (a -n \right )^{2} v -4 \left (a -n \right ) v^{2}-4 \left (a -n \right ) v +8 a \,v^{2}+8 a v +v^{4}+2 v^{3}-24 n a -34 a \left (a -n \right )+17 n \left (a -n \right )-2 n^{2} v^{2}-2 n^{2} v -8 v^{2} a^{2}-8 n^{3} a +60 n \,a^{2}-34 n \,a^{3}+25 n^{2} a^{2}+8 v^{2} n a -8 v \,a^{2}+8 v n a +12 \left (a -n \right )^{2}-40 a^{3}\right )}{4 \left (4-6 n +2 n^{2}-3 a \,n^{2}+13 a^{2}+\left (a -n \right )^{4}+6 \left (a -n \right )^{3}+6 a^{2} n \left (a -n \right )-2 a \,n^{2} \left (a -n \right )-6 a n \left (a -n \right )^{2}-18 a n \left (a -n \right )+a^{4}-4 a^{3} \left (a -n \right )+6 a^{2} \left (a -n \right )^{2}-4 a \left (a -n \right )^{3}+n^{2} \left (a -n \right )^{2}+2 n \left (a -n \right )^{3}+18 a^{2} \left (a -n \right )-18 a \left (a -n \right )^{2}+3 n^{2} \left (a -n \right )+9 n \left (a -n \right )^{2}-13 n a -26 a \left (a -n \right )+13 n \left (a -n \right )+9 n \,a^{2}-2 n \,a^{3}+n^{2} a^{2}+13 \left (a -n \right )^{2}-6 a^{3}\right )}\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 <- Legendre successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 29
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*(n+1-2*a)*diff(y(x),x)-(4*a*x^2*(a-n)-(x^2-1)*(2*a+(v-n)*(v+n+1)))/(x^2-1)^2*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \left (\operatorname {LegendreP}\left (v , n , x\right ) c_{1} +\operatorname {LegendreQ}\left (v , n , x\right ) c_{2} \right ) \left (x^{2}-1\right )^{a -\frac {n}{2}} \]
✓ Solution by Mathematica
Time used: 0.057 (sec). Leaf size: 34
DSolve[y''[x] == -(((4*a*(a - n)*x^2 - (2*a + (-n + v)*(1 + n + v))*(-1 + x^2))*y[x])/(-1 + x^2)^2) - (2*(1 - 2*a + n)*x*y'[x])/(-1 + x^2),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (x^2-1\right )^{a-\frac {n}{2}} (c_1 P_v^n(x)+c_2 Q_v^n(x)) \]