Internal problem ID [9700]
Internal file name [OUTPUT/8642_Monday_June_06_2022_04_35_20_AM_37056716/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1373.
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.
\[ \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}-1}+\frac {\left (-a^{2} \left (x^{2}-1\right )^{2}-n \left (n +1\right ) \left (x^{2}-1\right )-m^{2}\right ) y}{\left (x^{2}-1\right )^{2}}=0} \]
\[ \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 )+\left (2 x^{3}-2 x \right ) y^{\prime }+\left (-a^{2} x^{4}+\left (2 a^{2}-n^{2}-n \right ) x^{2}-a^{2}-m^{2}+n^{2}+n \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 (a^{2} x^{4}-2 a^{2} x^{2}+n^{2} x^{2}+n \,x^{2}+a^{2}+m^{2}-n^{2}-n \right ) y}{x^{4}-2 x^{2}+1}-\frac {2 x y^{\prime }}{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 x y^{\prime }}{x^{2}-1}-\frac {\left (a^{2} x^{4}-2 a^{2} x^{2}+n^{2} x^{2}+n \,x^{2}+a^{2}+m^{2}-n^{2}-n \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}{x^{2}-1}, P_{3}\left (x \right )=-\frac {a^{2} x^{4}-2 a^{2} x^{2}+n^{2} x^{2}+n \,x^{2}+a^{2}+m^{2}-n^{2}-n}{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}}}=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}}}=-\frac {m^{2}}{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^{2}-1\right ) \left (x^{4}-2 x^{2}+1\right )+2 y^{\prime } x \left (x^{4}-2 x^{2}+1\right )-\left (a^{2} x^{4}-2 a^{2} x^{2}+n^{2} x^{2}+n \,x^{2}+a^{2}+m^{2}-n^{2}-n \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 (2 u^{5}-10 u^{4}+16 u^{3}-8 u^{2}\right ) \left (\frac {d}{d u}y \left (u \right )\right )+\left (-a^{2} u^{6}+6 a^{2} u^{5}-12 a^{2} u^{4}-n^{2} u^{4}+8 a^{2} u^{3}+4 n^{2} u^{3}-n \,u^{4}-m^{2} u^{2}-4 n^{2} u^{2}+4 n \,u^{3}+2 m^{2} u -4 n \,u^{2}\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..6 \\ {} & {} & 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} \\ {} & {} & -2 a_{0} \left (m +2 r \right ) \left (-m +2 r \right ) u^{1+r}+\left (-2 a_{1} \left (2+m +2 r \right ) \left (2-m +2 r \right )+a_{0} \left (-m^{2}-4 n^{2}+12 r^{2}-4 n +4 r \right )\right ) u^{2+r}+\left (-2 a_{2} \left (4+m +2 r \right ) \left (4-m +2 r \right )+a_{1} \left (-m^{2}-4 n^{2}+12 r^{2}-4 n +28 r +16\right )+2 a_{0} \left (4 a^{2}+2 n^{2}-3 r^{2}+2 n -2 r \right )\right ) u^{3+r}+\left (-2 a_{3} \left (6+m +2 r \right ) \left (6-m +2 r \right )+a_{2} \left (-m^{2}-4 n^{2}+12 r^{2}-4 n +52 r +56\right )+2 a_{1} \left (4 a^{2}+2 n^{2}-3 r^{2}+2 n -8 r -5\right )-a_{0} \left (12 a^{2}+n^{2}-r^{2}+n -r \right )\right ) u^{4+r}+\left (-2 a_{4} \left (8+m +2 r \right ) \left (8-m +2 r \right )+a_{3} \left (-m^{2}-4 n^{2}+12 r^{2}-4 n +76 r +120\right )+2 a_{2} \left (4 a^{2}+2 n^{2}-3 r^{2}+2 n -14 r -16\right )-a_{1} \left (12 a^{2}+n^{2}-r^{2}+n -3 r -2\right )+6 a_{0} a^{2}\right ) u^{5+r}+\left (\moverset {\infty }{\munderset {k =6}{\sum }}\left (-2 a_{k -1} \left (2 k -2+m +2 r \right ) \left (2 k -2-m +2 r \right )+a_{k -2} \left (12 \left (k -2\right )^{2}+24 \left (k -2\right ) r -m^{2}-4 n^{2}+12 r^{2}+4 k -8-4 n +4 r \right )+2 a_{k -3} \left (4 a^{2}-3 \left (k -3\right )^{2}-6 \left (k -3\right ) r +2 n^{2}-3 r^{2}-2 k +6+2 n -2 r \right )-a_{k -4} \left (12 a^{2}-\left (k -4\right )^{2}-2 \left (k -4\right ) r +n^{2}-r^{2}-k +4+n -r \right )+6 a_{k -5} a^{2}-a_{k -6} a^{2}\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -2 \left (m +2 r \right ) \left (-m +2 r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-\frac {m}{2}, \frac {m}{2}\right \} \\ \bullet & {} & \textrm {The coefficients of each power of}\hspace {3pt} u \hspace {3pt}\textrm {must be 0}\hspace {3pt} \\ {} & {} & \left [-2 a_{1} \left (2+m +2 r \right ) \left (2-m +2 r \right )+a_{0} \left (-m^{2}-4 n^{2}+12 r^{2}-4 n +4 r \right )=0, -2 a_{2} \left (4+m +2 r \right ) \left (4-m +2 r \right )+a_{1} \left (-m^{2}-4 n^{2}+12 r^{2}-4 n +28 r +16\right )+2 a_{0} \left (4 a^{2}+2 n^{2}-3 r^{2}+2 n -2 r \right )=0, -2 a_{3} \left (6+m +2 r \right ) \left (6-m +2 r \right )+a_{2} \left (-m^{2}-4 n^{2}+12 r^{2}-4 n +52 r +56\right )+2 a_{1} \left (4 a^{2}+2 n^{2}-3 r^{2}+2 n -8 r -5\right )-a_{0} \left (12 a^{2}+n^{2}-r^{2}+n -r \right )=0, -2 a_{4} \left (8+m +2 r \right ) \left (8-m +2 r \right )+a_{3} \left (-m^{2}-4 n^{2}+12 r^{2}-4 n +76 r +120\right )+2 a_{2} \left (4 a^{2}+2 n^{2}-3 r^{2}+2 n -14 r -16\right )-a_{1} \left (12 a^{2}+n^{2}-r^{2}+n -3 r -2\right )+6 a_{0} a^{2}=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{1}=\frac {a_{0} \left (m^{2}+4 n^{2}-12 r^{2}+4 n -4 r \right )}{2 \left (m^{2}-4 r^{2}-8 r -4\right )}, a_{2}=-\frac {a_{0} \left (16 a^{2} m^{2}-64 a^{2} r^{2}-m^{4}+12 m^{2} r^{2}-16 n^{4}+64 n^{2} r^{2}-96 r^{4}-128 a^{2} r +24 m^{2} r -32 n^{3}+64 n^{2} r +64 n \,r^{2}-256 r^{3}-64 a^{2}+16 m^{2}+16 n^{2}+64 n r -192 r^{2}+32 n -32 r \right )}{4 \left (m^{4}-8 m^{2} r^{2}+16 r^{4}-24 m^{2} r +96 r^{3}-20 m^{2}+208 r^{2}+192 r +64\right )}, a_{3}=\frac {a_{0} \left (16 a^{2} m^{4}-128 a^{2} m^{2} n^{2}+128 a^{2} m^{2} r^{2}+512 a^{2} n^{2} r^{2}-768 a^{2} r^{4}+m^{6}-16 m^{4} r^{2}-16 m^{2} n^{4}+32 m^{2} n^{2} r^{2}+80 m^{2} r^{4}+64 n^{6}-320 n^{4} r^{2}+640 n^{2} r^{4}-640 r^{6}-128 a^{2} m^{2} n +128 a^{2} m^{2} r +1536 a^{2} n^{2} r +512 a^{2} n \,r^{2}-3584 a^{2} r^{3}-48 m^{4} r -32 m^{2} n^{3}+32 m^{2} n^{2} r +32 m^{2} n \,r^{2}+544 m^{2} r^{3}+192 n^{5}-576 n^{4} r -640 n^{3} r^{2}+2560 n^{2} r^{3}+640 n \,r^{4}-4480 r^{5}+256 a^{2} m^{2}+1280 a^{2} n^{2}+1536 a^{2} n r -5120 a^{2} r^{2}-52 m^{4}+16 m^{2} n^{2}+32 m^{2} n r +1328 m^{2} r^{2}-320 n^{4}-1152 n^{3} r +3392 n^{2} r^{2}+2560 n \,r^{3}-11520 r^{4}+1280 a^{2} n -2304 a^{2} r +32 m^{2} n +1376 m^{2} r -960 n^{3}+1728 n^{2} r +3712 n \,r^{2}-13184 r^{3}-512 a^{2}+576 m^{2}+256 n^{2}+2304 n r -6272 r^{2}+768 n -768 r \right )}{8 \left (m^{6}-12 m^{4} r^{2}+48 m^{2} r^{4}-64 r^{6}-48 m^{4} r +384 m^{2} r^{3}-768 r^{5}-56 m^{4}+1152 m^{2} r^{2}-3712 r^{4}+1536 m^{2} r -9216 r^{3}+784 m^{2}-12352 r^{2}-8448 r -2304\right )}, a_{4}=\frac {a_{0} \left (256 a^{4} m^{4}-2048 a^{4} m^{2} r^{2}+4096 a^{4} r^{4}+256 a^{2} m^{4} n^{2}-384 a^{2} m^{4} r^{2}-768 a^{2} m^{2} n^{4}+1024 a^{2} m^{2} n^{2} r^{2}+3072 a^{2} n^{4} r^{2}-8192 a^{2} n^{2} r^{4}+6144 a^{2} r^{6}+m^{8}-20 r^{2} m^{6}+160 r^{4} m^{4}-128 m^{2} n^{6}+576 m^{2} n^{4} r^{2}-768 m^{2} n^{2} r^{4}-320 r^{6} m^{2}+256 n^{8}-1536 n^{6} r^{2}+3840 n^{4} r^{4}-5120 n^{2} r^{6}+3840 r^{8}-8192 a^{4} m^{2} r +32768 a^{4} r^{3}+256 a^{2} m^{4} n -1024 a^{2} m^{4} r -1536 a^{2} m^{2} n^{3}+1024 a^{2} m^{2} n^{2} r +1024 a^{2} m^{2} n \,r^{2}+2048 a^{2} m^{2} r^{3}+12288 a^{2} n^{4} r +6144 a^{2} n^{3} r^{2}-53248 a^{2} n^{2} r^{3}-8192 a^{2} n \,r^{4}+57344 a^{2} r^{5}-80 r \,m^{6}+1280 r^{3} m^{4}-384 m^{2} n^{5}+1536 m^{2} n^{4} r +1152 m^{2} n^{3} r^{2}-3840 m^{2} n^{2} r^{3}-768 m^{2} n \,r^{4}-5376 r^{5} m^{2}+1024 n^{7}-4096 n^{6} r -4608 n^{5} r^{2}+21504 n^{4} r^{3}+7680 n^{3} r^{4}-46080 n^{2} r^{5}-5120 n \,r^{6}+51200 r^{7}-10240 a^{4} m^{2}+90112 a^{4} r^{2}-1664 a^{2} m^{4}+2304 a^{2} m^{2} n^{2}+1024 a^{2} m^{2} n r +1792 a^{2} m^{2} r^{2}+14336 a^{2} n^{4}+24576 a^{2} n^{3} r -115712 a^{2} n^{2} r^{2}-53248 a^{2} n \,r^{3}+199680 a^{2} r^{4}-116 m^{6}+4080 m^{4} r^{2}+1600 m^{2} n^{4}+3072 m^{2} n^{3} r -6720 m^{2} n^{2} r^{2}-3840 m^{2} n \,r^{3}-32640 m^{2} r^{4}-3584 n^{6}-12288 n^{5} r +43776 n^{4} r^{2}+43008 n^{3} r^{3}-159488 n^{2} r^{4}-46080 n \,r^{5}+276480 r^{6}+98304 a^{4} r +3072 a^{2} m^{2} n -6656 a^{2} m^{2} r +28672 a^{2} n^{3}-94208 a^{2} n^{2} r -118784 a^{2} n \,r^{2}+325632 a^{2} r^{3}+6080 m^{4} r +3840 m^{2} n^{3}-4608 m^{2} n^{2} r -7296 m^{2} n \,r^{2}-95360 m^{2} r^{3}-14336 n^{5}+38912 n^{4} r +95232 n^{3} r^{2}-267264 n^{2} r^{3}-163328 n \,r^{4}+774656 r^{5}+36864 a^{4}-9472 a^{2} m^{2}-30720 a^{2} n^{2}-106496 a^{2} n r +263168 a^{2} r^{2}+3904 m^{4}-1472 m^{2} n^{2}-6144 m^{2} n r -144960 m^{2} r^{2}+12544 n^{4}+98304 n^{3} r -222464 n^{2} r^{2}-288768 n \,r^{3}+1194240 r^{4}-45056 a^{2} n +106496 a^{2} r -3456 m^{2} n -111744 m^{2} r +50176 n^{3}-83968 n^{2} r -270848 n \,r^{2}+980480 r^{3}+24576 a^{2}-36864 m^{2}-9216 n^{2}-135168 n r +368640 r^{2}-36864 n +36864 r \right )}{16 \left (m^{8}-16 r^{2} m^{6}+96 r^{4} m^{4}-256 r^{6} m^{2}+256 r^{8}-80 r \,m^{6}+960 r^{3} m^{4}-3840 r^{5} m^{2}+5120 r^{7}-120 m^{6}+3680 m^{4} r^{2}-23680 m^{2} r^{4}+43520 r^{6}+6400 m^{4} r -76800 m^{2} r^{3}+204800 r^{5}+4368 m^{4}-138368 m^{2} r^{2}+581888 r^{4}-131840 m^{2} r +1018880 r^{3}-52480 m^{2}+1070080 r^{2}+614400 r +147456\right )}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (12 a_{k -2}-8 a_{k -1}+a_{k -4}-6 a_{k -3}\right ) k^{2}+\left (2 \left (12 a_{k -2}-8 a_{k -1}+a_{k -4}-6 a_{k -3}\right ) r -44 a_{k -2}+16 a_{k -1}-7 a_{k -4}+32 a_{k -3}\right ) k +\left (12 a_{k -2}-8 a_{k -1}+a_{k -4}-6 a_{k -3}\right ) r^{2}+\left (-44 a_{k -2}+16 a_{k -1}-7 a_{k -4}+32 a_{k -3}\right ) r +\left (-12 a^{2}-n^{2}-n +12\right ) a_{k -4}+2 \left (4 a^{2}+2 n^{2}+2 n -21\right ) a_{k -3}+\left (-m^{2}-4 n^{2}-4 n +40\right ) a_{k -2}+2 \left (m^{2}-4\right ) a_{k -1}-a^{2} \left (a_{k -6}-6 a_{k -5}\right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +6 \\ {} & {} & \left (12 a_{k +4}-8 a_{k +5}+a_{k +2}-6 a_{k +3}\right ) \left (k +6\right )^{2}+\left (2 \left (12 a_{k +4}-8 a_{k +5}+a_{k +2}-6 a_{k +3}\right ) r -44 a_{k +4}+16 a_{k +5}-7 a_{k +2}+32 a_{k +3}\right ) \left (k +6\right )+\left (12 a_{k +4}-8 a_{k +5}+a_{k +2}-6 a_{k +3}\right ) r^{2}+\left (-44 a_{k +4}+16 a_{k +5}-7 a_{k +2}+32 a_{k +3}\right ) r +\left (-12 a^{2}-n^{2}-n +12\right ) a_{k +2}+2 \left (4 a^{2}+2 n^{2}+2 n -21\right ) a_{k +3}+\left (-m^{2}-4 n^{2}-4 n +40\right ) a_{k +4}+2 \left (m^{2}-4\right ) a_{k +5}-a^{2} \left (a_{k}-6 a_{k +1}\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +5}=-\frac {a_{k} a^{2}-6 a^{2} a_{k +1}+12 a^{2} a_{k +2}-8 a^{2} a_{k +3}-k^{2} a_{k +2}+6 k^{2} a_{k +3}-12 k^{2} a_{k +4}-2 k r a_{k +2}+12 k r a_{k +3}-24 k r a_{k +4}+m^{2} a_{k +4}+n^{2} a_{k +2}-4 n^{2} a_{k +3}+4 n^{2} a_{k +4}-r^{2} a_{k +2}+6 r^{2} a_{k +3}-12 r^{2} a_{k +4}-5 k a_{k +2}+40 k a_{k +3}-100 k a_{k +4}+n a_{k +2}-4 n a_{k +3}+4 n a_{k +4}-5 r a_{k +2}+40 r a_{k +3}-100 r a_{k +4}-6 a_{k +2}+66 a_{k +3}-208 a_{k +4}}{2 \left (4 k^{2}+8 k r -m^{2}+4 r^{2}+40 k +40 r +100\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {m}{2} \\ {} & {} & a_{k +5}=-\frac {k m a_{k +2}-6 k m a_{k +3}+12 k m a_{k +4}-208 a_{k +4}-6 a_{k +2}+66 a_{k +3}-2 m^{2} a_{k +4}+n^{2} a_{k +2}-4 n^{2} a_{k +3}+4 n^{2} a_{k +4}-5 k a_{k +2}+40 k a_{k +3}-100 k a_{k +4}+n a_{k +2}-4 n a_{k +3}+4 n a_{k +4}-\frac {1}{4} m^{2} a_{k +2}+\frac {3}{2} m^{2} a_{k +3}+\frac {5}{2} m a_{k +2}-20 m a_{k +3}+50 m a_{k +4}-6 a^{2} a_{k +1}+12 a^{2} a_{k +2}-8 a^{2} a_{k +3}-k^{2} a_{k +2}+6 k^{2} a_{k +3}-12 k^{2} a_{k +4}+a_{k} a^{2}}{2 \left (4 k^{2}-4 k m +40 k -20 m +100\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {m}{2} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k -\frac {m}{2}}, a_{k +5}=-\frac {k m a_{k +2}-6 k m a_{k +3}+12 k m a_{k +4}-208 a_{k +4}-6 a_{k +2}+66 a_{k +3}-2 m^{2} a_{k +4}+n^{2} a_{k +2}-4 n^{2} a_{k +3}+4 n^{2} a_{k +4}-5 k a_{k +2}+40 k a_{k +3}-100 k a_{k +4}+n a_{k +2}-4 n a_{k +3}+4 n a_{k +4}-\frac {1}{4} m^{2} a_{k +2}+\frac {3}{2} m^{2} a_{k +3}+\frac {5}{2} m a_{k +2}-20 m a_{k +3}+50 m a_{k +4}-6 a^{2} a_{k +1}+12 a^{2} a_{k +2}-8 a^{2} a_{k +3}-k^{2} a_{k +2}+6 k^{2} a_{k +3}-12 k^{2} a_{k +4}+a_{k} a^{2}}{2 \left (4 k^{2}-4 k m +40 k -20 m +100\right )}, a_{1}=\frac {a_{0} \left (-2 m^{2}+4 n^{2}+2 m +4 n \right )}{2 \left (4 m -4\right )}, a_{2}=-\frac {a_{0} \left (-4 m^{4}+16 m^{2} n^{2}-16 n^{4}+64 a^{2} m +20 m^{3}+16 m^{2} n -32 m \,n^{2}-32 n^{3}-64 a^{2}-32 m^{2}-32 m n +16 n^{2}+16 m +32 n \right )}{4 \left (32 m^{2}-96 m +64\right )}, a_{3}=\frac {a_{0} \left (-8 m^{6}+48 m^{4} n^{2}-96 m^{2} n^{4}+64 n^{6}+384 a^{2} m^{3}-768 a^{2} m \,n^{2}+96 m^{5}+48 m^{4} n -336 m^{3} n^{2}-192 m^{2} n^{3}+288 m \,n^{4}+192 n^{5}-1024 a^{2} m^{2}-768 a^{2} m n +1280 a^{2} n^{2}-440 m^{4}-336 m^{3} n +864 m^{2} n^{2}+576 m \,n^{3}-320 n^{4}+1152 a^{2} m +1280 a^{2} n +960 m^{3}+960 m^{2} n -864 m \,n^{2}-960 n^{3}-512 a^{2}-992 m^{2}-1152 m n +256 n^{2}+384 m +768 n \right )}{8 \left (384 m^{3}-2304 m^{2}+4224 m -2304\right )}, a_{4}=\frac {a_{0} \left (16 m^{8}-128 m^{6} n^{2}+384 m^{4} n^{4}-512 m^{2} n^{6}+256 n^{8}-1536 a^{2} m^{5}+6144 a^{2} m^{3} n^{2}-6144 a^{2} m \,n^{4}-352 m^{7}-128 m^{6} n +1920 m^{5} n^{2}+768 m^{4} n^{3}-3456 m^{3} n^{4}-1536 m^{2} n^{5}+2048 m \,n^{6}+1024 n^{7}+12288 a^{4} m^{2}+11264 a^{2} m^{4}+6144 a^{2} m^{3} n -26624 a^{2} m^{2} n^{2}-12288 a^{2} m \,n^{3}+14336 a^{2} n^{4}+3184 m^{6}+1920 m^{5} n -11648 m^{4} n^{2}-6912 m^{3} n^{3}+12544 m^{2} n^{4}+6144 m \,n^{5}-3584 n^{6}-49152 a^{4} m -37376 a^{2} m^{3}-26624 a^{2} m^{2} n +47104 a^{2} m \,n^{2}+28672 a^{2} n^{3}-15328 m^{5}-12032 m^{4} n +35712 m^{3} n^{2}+27648 m^{2} n^{3}-19456 m \,n^{4}-14336 n^{5}+36864 a^{4}+56320 a^{2} m^{2}+53248 a^{2} m n -30720 a^{2} n^{2}+42304 m^{4}+39168 m^{3} n -57088 m^{2} n^{2}-49152 m \,n^{3}+12544 n^{4}-53248 a^{2} m -45056 a^{2} n -66688 m^{3}-71168 m^{2} n +41984 m \,n^{2}+50176 n^{3}+24576 a^{2}+55296 m^{2}+67584 m n -9216 n^{2}-18432 m -36864 n \right )}{16 \left (6144 m^{4}-61440 m^{3}+215040 m^{2}-307200 m +147456\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 -\frac {m}{2}}, a_{k +5}=-\frac {k m a_{k +2}-6 k m a_{k +3}+12 k m a_{k +4}-208 a_{k +4}-6 a_{k +2}+66 a_{k +3}-2 m^{2} a_{k +4}+n^{2} a_{k +2}-4 n^{2} a_{k +3}+4 n^{2} a_{k +4}-5 k a_{k +2}+40 k a_{k +3}-100 k a_{k +4}+n a_{k +2}-4 n a_{k +3}+4 n a_{k +4}-\frac {1}{4} m^{2} a_{k +2}+\frac {3}{2} m^{2} a_{k +3}+\frac {5}{2} m a_{k +2}-20 m a_{k +3}+50 m a_{k +4}-6 a^{2} a_{k +1}+12 a^{2} a_{k +2}-8 a^{2} a_{k +3}-k^{2} a_{k +2}+6 k^{2} a_{k +3}-12 k^{2} a_{k +4}+a_{k} a^{2}}{2 \left (4 k^{2}-4 k m +40 k -20 m +100\right )}, a_{1}=\frac {a_{0} \left (-2 m^{2}+4 n^{2}+2 m +4 n \right )}{2 \left (4 m -4\right )}, a_{2}=-\frac {a_{0} \left (-4 m^{4}+16 m^{2} n^{2}-16 n^{4}+64 a^{2} m +20 m^{3}+16 m^{2} n -32 m \,n^{2}-32 n^{3}-64 a^{2}-32 m^{2}-32 m n +16 n^{2}+16 m +32 n \right )}{4 \left (32 m^{2}-96 m +64\right )}, a_{3}=\frac {a_{0} \left (-8 m^{6}+48 m^{4} n^{2}-96 m^{2} n^{4}+64 n^{6}+384 a^{2} m^{3}-768 a^{2} m \,n^{2}+96 m^{5}+48 m^{4} n -336 m^{3} n^{2}-192 m^{2} n^{3}+288 m \,n^{4}+192 n^{5}-1024 a^{2} m^{2}-768 a^{2} m n +1280 a^{2} n^{2}-440 m^{4}-336 m^{3} n +864 m^{2} n^{2}+576 m \,n^{3}-320 n^{4}+1152 a^{2} m +1280 a^{2} n +960 m^{3}+960 m^{2} n -864 m \,n^{2}-960 n^{3}-512 a^{2}-992 m^{2}-1152 m n +256 n^{2}+384 m +768 n \right )}{8 \left (384 m^{3}-2304 m^{2}+4224 m -2304\right )}, a_{4}=\frac {a_{0} \left (16 m^{8}-128 m^{6} n^{2}+384 m^{4} n^{4}-512 m^{2} n^{6}+256 n^{8}-1536 a^{2} m^{5}+6144 a^{2} m^{3} n^{2}-6144 a^{2} m \,n^{4}-352 m^{7}-128 m^{6} n +1920 m^{5} n^{2}+768 m^{4} n^{3}-3456 m^{3} n^{4}-1536 m^{2} n^{5}+2048 m \,n^{6}+1024 n^{7}+12288 a^{4} m^{2}+11264 a^{2} m^{4}+6144 a^{2} m^{3} n -26624 a^{2} m^{2} n^{2}-12288 a^{2} m \,n^{3}+14336 a^{2} n^{4}+3184 m^{6}+1920 m^{5} n -11648 m^{4} n^{2}-6912 m^{3} n^{3}+12544 m^{2} n^{4}+6144 m \,n^{5}-3584 n^{6}-49152 a^{4} m -37376 a^{2} m^{3}-26624 a^{2} m^{2} n +47104 a^{2} m \,n^{2}+28672 a^{2} n^{3}-15328 m^{5}-12032 m^{4} n +35712 m^{3} n^{2}+27648 m^{2} n^{3}-19456 m \,n^{4}-14336 n^{5}+36864 a^{4}+56320 a^{2} m^{2}+53248 a^{2} m n -30720 a^{2} n^{2}+42304 m^{4}+39168 m^{3} n -57088 m^{2} n^{2}-49152 m \,n^{3}+12544 n^{4}-53248 a^{2} m -45056 a^{2} n -66688 m^{3}-71168 m^{2} n +41984 m \,n^{2}+50176 n^{3}+24576 a^{2}+55296 m^{2}+67584 m n -9216 n^{2}-18432 m -36864 n \right )}{16 \left (6144 m^{4}-61440 m^{3}+215040 m^{2}-307200 m +147456\right )}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {m}{2} \\ {} & {} & a_{k +5}=-\frac {-k m a_{k +2}+6 k m a_{k +3}-12 k m a_{k +4}-208 a_{k +4}-6 a_{k +2}+66 a_{k +3}-2 m^{2} a_{k +4}+n^{2} a_{k +2}-4 n^{2} a_{k +3}+4 n^{2} a_{k +4}-5 k a_{k +2}+40 k a_{k +3}-100 k a_{k +4}+n a_{k +2}-4 n a_{k +3}+4 n a_{k +4}-\frac {1}{4} m^{2} a_{k +2}+\frac {3}{2} m^{2} a_{k +3}-\frac {5}{2} m a_{k +2}+20 m a_{k +3}-50 m a_{k +4}-6 a^{2} a_{k +1}+12 a^{2} a_{k +2}-8 a^{2} a_{k +3}-k^{2} a_{k +2}+6 k^{2} a_{k +3}-12 k^{2} a_{k +4}+a_{k} a^{2}}{2 \left (4 k^{2}+4 k m +40 k +20 m +100\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {m}{2} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +\frac {m}{2}}, a_{k +5}=-\frac {-k m a_{k +2}+6 k m a_{k +3}-12 k m a_{k +4}-208 a_{k +4}-6 a_{k +2}+66 a_{k +3}-2 m^{2} a_{k +4}+n^{2} a_{k +2}-4 n^{2} a_{k +3}+4 n^{2} a_{k +4}-5 k a_{k +2}+40 k a_{k +3}-100 k a_{k +4}+n a_{k +2}-4 n a_{k +3}+4 n a_{k +4}-\frac {1}{4} m^{2} a_{k +2}+\frac {3}{2} m^{2} a_{k +3}-\frac {5}{2} m a_{k +2}+20 m a_{k +3}-50 m a_{k +4}-6 a^{2} a_{k +1}+12 a^{2} a_{k +2}-8 a^{2} a_{k +3}-k^{2} a_{k +2}+6 k^{2} a_{k +3}-12 k^{2} a_{k +4}+a_{k} a^{2}}{2 \left (4 k^{2}+4 k m +40 k +20 m +100\right )}, a_{1}=\frac {a_{0} \left (-2 m^{2}+4 n^{2}-2 m +4 n \right )}{2 \left (-4 m -4\right )}, a_{2}=-\frac {a_{0} \left (-4 m^{4}+16 m^{2} n^{2}-16 n^{4}-64 a^{2} m -20 m^{3}+16 m^{2} n +32 m \,n^{2}-32 n^{3}-64 a^{2}-32 m^{2}+32 m n +16 n^{2}-16 m +32 n \right )}{4 \left (32 m^{2}+96 m +64\right )}, a_{3}=\frac {a_{0} \left (-8 m^{6}+48 m^{4} n^{2}-96 m^{2} n^{4}+64 n^{6}-384 a^{2} m^{3}+768 a^{2} m \,n^{2}-96 m^{5}+48 m^{4} n +336 m^{3} n^{2}-192 m^{2} n^{3}-288 m \,n^{4}+192 n^{5}-1024 a^{2} m^{2}+768 a^{2} m n +1280 a^{2} n^{2}-440 m^{4}+336 m^{3} n +864 m^{2} n^{2}-576 m \,n^{3}-320 n^{4}-1152 a^{2} m +1280 a^{2} n -960 m^{3}+960 m^{2} n +864 m \,n^{2}-960 n^{3}-512 a^{2}-992 m^{2}+1152 m n +256 n^{2}-384 m +768 n \right )}{8 \left (-384 m^{3}-2304 m^{2}-4224 m -2304\right )}, a_{4}=\frac {a_{0} \left (16 m^{8}-128 m^{6} n^{2}+384 m^{4} n^{4}-512 m^{2} n^{6}+256 n^{8}+1536 a^{2} m^{5}-6144 a^{2} m^{3} n^{2}+6144 a^{2} m \,n^{4}+352 m^{7}-128 m^{6} n -1920 m^{5} n^{2}+768 m^{4} n^{3}+3456 m^{3} n^{4}-1536 m^{2} n^{5}-2048 m \,n^{6}+1024 n^{7}+12288 a^{4} m^{2}+11264 a^{2} m^{4}-6144 a^{2} m^{3} n -26624 a^{2} m^{2} n^{2}+12288 a^{2} m \,n^{3}+14336 a^{2} n^{4}+3184 m^{6}-1920 m^{5} n -11648 m^{4} n^{2}+6912 m^{3} n^{3}+12544 m^{2} n^{4}-6144 m \,n^{5}-3584 n^{6}+49152 a^{4} m +37376 a^{2} m^{3}-26624 a^{2} m^{2} n -47104 a^{2} m \,n^{2}+28672 a^{2} n^{3}+15328 m^{5}-12032 m^{4} n -35712 m^{3} n^{2}+27648 m^{2} n^{3}+19456 m \,n^{4}-14336 n^{5}+36864 a^{4}+56320 a^{2} m^{2}-53248 a^{2} m n -30720 a^{2} n^{2}+42304 m^{4}-39168 m^{3} n -57088 m^{2} n^{2}+49152 m \,n^{3}+12544 n^{4}+53248 a^{2} m -45056 a^{2} n +66688 m^{3}-71168 m^{2} n -41984 m \,n^{2}+50176 n^{3}+24576 a^{2}+55296 m^{2}-67584 m n -9216 n^{2}+18432 m -36864 n \right )}{16 \left (6144 m^{4}+61440 m^{3}+215040 m^{2}+307200 m +147456\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 +\frac {m}{2}}, a_{k +5}=-\frac {-k m a_{k +2}+6 k m a_{k +3}-12 k m a_{k +4}-208 a_{k +4}-6 a_{k +2}+66 a_{k +3}-2 m^{2} a_{k +4}+n^{2} a_{k +2}-4 n^{2} a_{k +3}+4 n^{2} a_{k +4}-5 k a_{k +2}+40 k a_{k +3}-100 k a_{k +4}+n a_{k +2}-4 n a_{k +3}+4 n a_{k +4}-\frac {1}{4} m^{2} a_{k +2}+\frac {3}{2} m^{2} a_{k +3}-\frac {5}{2} m a_{k +2}+20 m a_{k +3}-50 m a_{k +4}-6 a^{2} a_{k +1}+12 a^{2} a_{k +2}-8 a^{2} a_{k +3}-k^{2} a_{k +2}+6 k^{2} a_{k +3}-12 k^{2} a_{k +4}+a_{k} a^{2}}{2 \left (4 k^{2}+4 k m +40 k +20 m +100\right )}, a_{1}=\frac {a_{0} \left (-2 m^{2}+4 n^{2}-2 m +4 n \right )}{2 \left (-4 m -4\right )}, a_{2}=-\frac {a_{0} \left (-4 m^{4}+16 m^{2} n^{2}-16 n^{4}-64 a^{2} m -20 m^{3}+16 m^{2} n +32 m \,n^{2}-32 n^{3}-64 a^{2}-32 m^{2}+32 m n +16 n^{2}-16 m +32 n \right )}{4 \left (32 m^{2}+96 m +64\right )}, a_{3}=\frac {a_{0} \left (-8 m^{6}+48 m^{4} n^{2}-96 m^{2} n^{4}+64 n^{6}-384 a^{2} m^{3}+768 a^{2} m \,n^{2}-96 m^{5}+48 m^{4} n +336 m^{3} n^{2}-192 m^{2} n^{3}-288 m \,n^{4}+192 n^{5}-1024 a^{2} m^{2}+768 a^{2} m n +1280 a^{2} n^{2}-440 m^{4}+336 m^{3} n +864 m^{2} n^{2}-576 m \,n^{3}-320 n^{4}-1152 a^{2} m +1280 a^{2} n -960 m^{3}+960 m^{2} n +864 m \,n^{2}-960 n^{3}-512 a^{2}-992 m^{2}+1152 m n +256 n^{2}-384 m +768 n \right )}{8 \left (-384 m^{3}-2304 m^{2}-4224 m -2304\right )}, a_{4}=\frac {a_{0} \left (16 m^{8}-128 m^{6} n^{2}+384 m^{4} n^{4}-512 m^{2} n^{6}+256 n^{8}+1536 a^{2} m^{5}-6144 a^{2} m^{3} n^{2}+6144 a^{2} m \,n^{4}+352 m^{7}-128 m^{6} n -1920 m^{5} n^{2}+768 m^{4} n^{3}+3456 m^{3} n^{4}-1536 m^{2} n^{5}-2048 m \,n^{6}+1024 n^{7}+12288 a^{4} m^{2}+11264 a^{2} m^{4}-6144 a^{2} m^{3} n -26624 a^{2} m^{2} n^{2}+12288 a^{2} m \,n^{3}+14336 a^{2} n^{4}+3184 m^{6}-1920 m^{5} n -11648 m^{4} n^{2}+6912 m^{3} n^{3}+12544 m^{2} n^{4}-6144 m \,n^{5}-3584 n^{6}+49152 a^{4} m +37376 a^{2} m^{3}-26624 a^{2} m^{2} n -47104 a^{2} m \,n^{2}+28672 a^{2} n^{3}+15328 m^{5}-12032 m^{4} n -35712 m^{3} n^{2}+27648 m^{2} n^{3}+19456 m \,n^{4}-14336 n^{5}+36864 a^{4}+56320 a^{2} m^{2}-53248 a^{2} m n -30720 a^{2} n^{2}+42304 m^{4}-39168 m^{3} n -57088 m^{2} n^{2}+49152 m \,n^{3}+12544 n^{4}+53248 a^{2} m -45056 a^{2} n +66688 m^{3}-71168 m^{2} n -41984 m \,n^{2}+50176 n^{3}+24576 a^{2}+55296 m^{2}-67584 m n -9216 n^{2}+18432 m -36864 n \right )}{16 \left (6144 m^{4}+61440 m^{3}+215040 m^{2}+307200 m +147456\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 -\frac {m}{2}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}c_{k} \left (x +1\right )^{k +\frac {m}{2}}\right ), b_{k +5}=-\frac {-2 m^{2} b_{k +4}+n^{2} b_{k +2}-4 n^{2} b_{k +3}+4 n^{2} b_{k +4}+40 k b_{k +3}-100 k b_{k +4}+n b_{k +2}-4 n b_{k +3}+4 n b_{k +4}-\frac {1}{4} m^{2} b_{k +2}+\frac {3}{2} m^{2} b_{k +3}+\frac {5}{2} m b_{k +2}-20 m b_{k +3}+50 m b_{k +4}-8 a^{2} b_{k +3}+6 k^{2} b_{k +3}-12 k^{2} b_{k +4}+66 b_{k +3}+k m b_{k +2}-6 k m b_{k +3}+12 k m b_{k +4}-6 b_{k +2}+b_{k} a^{2}-6 a^{2} b_{k +1}+12 a^{2} b_{k +2}-k^{2} b_{k +2}-5 k b_{k +2}-208 b_{k +4}}{2 \left (4 k^{2}-4 k m +40 k -20 m +100\right )}, b_{1}=\frac {b_{0} \left (-2 m^{2}+4 n^{2}+2 m +4 n \right )}{2 \left (4 m -4\right )}, b_{2}=-\frac {b_{0} \left (-4 m^{4}+16 m^{2} n^{2}-16 n^{4}+64 a^{2} m +20 m^{3}+16 m^{2} n -32 m \,n^{2}-32 n^{3}-64 a^{2}-32 m^{2}-32 m n +16 n^{2}+16 m +32 n \right )}{4 \left (32 m^{2}-96 m +64\right )}, b_{3}=\frac {b_{0} \left (-8 m^{6}+48 m^{4} n^{2}-96 m^{2} n^{4}+64 n^{6}+384 a^{2} m^{3}-768 a^{2} m \,n^{2}+96 m^{5}+48 m^{4} n -336 m^{3} n^{2}-192 m^{2} n^{3}+288 m \,n^{4}+192 n^{5}-1024 a^{2} m^{2}-768 a^{2} m n +1280 a^{2} n^{2}-440 m^{4}-336 m^{3} n +864 m^{2} n^{2}+576 m \,n^{3}-320 n^{4}+1152 a^{2} m +1280 a^{2} n +960 m^{3}+960 m^{2} n -864 m \,n^{2}-960 n^{3}-512 a^{2}-992 m^{2}-1152 m n +256 n^{2}+384 m +768 n \right )}{8 \left (384 m^{3}-2304 m^{2}+4224 m -2304\right )}, b_{4}=\frac {b_{0} \left (16 m^{8}-128 m^{6} n^{2}+384 m^{4} n^{4}-512 m^{2} n^{6}+256 n^{8}-1536 a^{2} m^{5}+6144 a^{2} m^{3} n^{2}-6144 a^{2} m \,n^{4}-352 m^{7}-128 m^{6} n +1920 m^{5} n^{2}+768 m^{4} n^{3}-3456 m^{3} n^{4}-1536 m^{2} n^{5}+2048 m \,n^{6}+1024 n^{7}+12288 a^{4} m^{2}+11264 a^{2} m^{4}+6144 a^{2} m^{3} n -26624 a^{2} m^{2} n^{2}-12288 a^{2} m \,n^{3}+14336 a^{2} n^{4}+3184 m^{6}+1920 m^{5} n -11648 m^{4} n^{2}-6912 m^{3} n^{3}+12544 m^{2} n^{4}+6144 m \,n^{5}-3584 n^{6}-49152 a^{4} m -37376 a^{2} m^{3}-26624 a^{2} m^{2} n +47104 a^{2} m \,n^{2}+28672 a^{2} n^{3}-15328 m^{5}-12032 m^{4} n +35712 m^{3} n^{2}+27648 m^{2} n^{3}-19456 m \,n^{4}-14336 n^{5}+36864 a^{4}+56320 a^{2} m^{2}+53248 a^{2} m n -30720 a^{2} n^{2}+42304 m^{4}+39168 m^{3} n -57088 m^{2} n^{2}-49152 m \,n^{3}+12544 n^{4}-53248 a^{2} m -45056 a^{2} n -66688 m^{3}-71168 m^{2} n +41984 m \,n^{2}+50176 n^{3}+24576 a^{2}+55296 m^{2}+67584 m n -9216 n^{2}-18432 m -36864 n \right )}{16 \left (6144 m^{4}-61440 m^{3}+215040 m^{2}-307200 m +147456\right )}, c_{k +5}=-\frac {-2 m^{2} c_{k +4}+n^{2} c_{k +2}-4 n^{2} c_{k +3}+4 n^{2} c_{k +4}+40 k c_{k +3}-100 k c_{k +4}+n c_{k +2}-4 n c_{k +3}+4 n c_{k +4}-\frac {1}{4} m^{2} c_{k +2}+\frac {3}{2} m^{2} c_{k +3}-\frac {5}{2} m c_{k +2}+20 m c_{k +3}-50 m c_{k +4}-8 a^{2} c_{k +3}+6 k^{2} c_{k +3}-12 k^{2} c_{k +4}+66 c_{k +3}-k m c_{k +2}+6 k m c_{k +3}-12 k m c_{k +4}-6 c_{k +2}+c_{k} a^{2}-6 a^{2} c_{k +1}+12 a^{2} c_{k +2}-k^{2} c_{k +2}-5 k c_{k +2}-208 c_{k +4}}{2 \left (4 k^{2}+4 k m +40 k +20 m +100\right )}, c_{1}=\frac {c_{0} \left (-2 m^{2}+4 n^{2}-2 m +4 n \right )}{2 \left (-4 m -4\right )}, c_{2}=-\frac {c_{0} \left (-4 m^{4}+16 m^{2} n^{2}-16 n^{4}-64 a^{2} m -20 m^{3}+16 m^{2} n +32 m \,n^{2}-32 n^{3}-64 a^{2}-32 m^{2}+32 m n +16 n^{2}-16 m +32 n \right )}{4 \left (32 m^{2}+96 m +64\right )}, c_{3}=\frac {c_{0} \left (-8 m^{6}+48 m^{4} n^{2}-96 m^{2} n^{4}+64 n^{6}-384 a^{2} m^{3}+768 a^{2} m \,n^{2}-96 m^{5}+48 m^{4} n +336 m^{3} n^{2}-192 m^{2} n^{3}-288 m \,n^{4}+192 n^{5}-1024 a^{2} m^{2}+768 a^{2} m n +1280 a^{2} n^{2}-440 m^{4}+336 m^{3} n +864 m^{2} n^{2}-576 m \,n^{3}-320 n^{4}-1152 a^{2} m +1280 a^{2} n -960 m^{3}+960 m^{2} n +864 m \,n^{2}-960 n^{3}-512 a^{2}-992 m^{2}+1152 m n +256 n^{2}-384 m +768 n \right )}{8 \left (-384 m^{3}-2304 m^{2}-4224 m -2304\right )}, c_{4}=\frac {c_{0} \left (16 m^{8}-128 m^{6} n^{2}+384 m^{4} n^{4}-512 m^{2} n^{6}+256 n^{8}+1536 a^{2} m^{5}-6144 a^{2} m^{3} n^{2}+6144 a^{2} m \,n^{4}+352 m^{7}-128 m^{6} n -1920 m^{5} n^{2}+768 m^{4} n^{3}+3456 m^{3} n^{4}-1536 m^{2} n^{5}-2048 m \,n^{6}+1024 n^{7}+12288 a^{4} m^{2}+11264 a^{2} m^{4}-6144 a^{2} m^{3} n -26624 a^{2} m^{2} n^{2}+12288 a^{2} m \,n^{3}+14336 a^{2} n^{4}+3184 m^{6}-1920 m^{5} n -11648 m^{4} n^{2}+6912 m^{3} n^{3}+12544 m^{2} n^{4}-6144 m \,n^{5}-3584 n^{6}+49152 a^{4} m +37376 a^{2} m^{3}-26624 a^{2} m^{2} n -47104 a^{2} m \,n^{2}+28672 a^{2} n^{3}+15328 m^{5}-12032 m^{4} n -35712 m^{3} n^{2}+27648 m^{2} n^{3}+19456 m \,n^{4}-14336 n^{5}+36864 a^{4}+56320 a^{2} m^{2}-53248 a^{2} m n -30720 a^{2} n^{2}+42304 m^{4}-39168 m^{3} n -57088 m^{2} n^{2}+49152 m \,n^{3}+12544 n^{4}+53248 a^{2} m -45056 a^{2} n +66688 m^{3}-71168 m^{2} n -41984 m \,n^{2}+50176 n^{3}+24576 a^{2}+55296 m^{2}-67584 m n -9216 n^{2}+18432 m -36864 n \right )}{16 \left (6144 m^{4}+61440 m^{3}+215040 m^{2}+307200 m +147456\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 -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunC ODE, case a <> 0, e <> 0, c = 0 `
✓ Solution by Maple
Time used: 0.578 (sec). Leaf size: 84
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)-(-a^2*(x^2-1)^2-n*(n+1)*(x^2-1)-m^2)/(x^2-1)^2*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \left (\operatorname {HeunC}\left (0, \frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , x^{2}\right ) c_{2} x +\operatorname {HeunC}\left (0, -\frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , x^{2}\right ) c_{1} \right ) \left (x^{2}-1\right )^{\frac {m}{2}} \]
✓ Solution by Mathematica
Time used: 2.094 (sec). Leaf size: 103
DSolve[y''[x] == -(((-m^2 - n*(1 + n)*(-1 + x^2) - a^2*(-1 + x^2)^2)*y[x])/(-1 + x^2)^2) - (2*x*y'[x])/(-1 + x^2),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (x^2-1\right )^{m/2} \left (c_1 \text {HeunC}\left [\frac {1}{4} \left (-a^2-m (m+1)+n^2+n\right ),-\frac {a^2}{4},\frac {1}{2},m+1,0,x^2\right ]+c_2 x \text {HeunC}\left [\frac {1}{4} \left (-a^2-(m-n+1) (m+n+2)\right ),-\frac {a^2}{4},\frac {3}{2},m+1,0,x^2\right ]\right ) \]