Internal problem ID [11059]
Internal file name [OUTPUT/10316_Wednesday_January_24_2024_10_07_08_PM_15066980/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-7 Equation of form
\((a_4 x^4+a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 235.
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.
\[ \boxed {\left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }+\left (\left (x^{2}-1\right ) \left (a^{2} x^{2}-\lambda \right )-m^{2}\right ) y=0} \]
\[ \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 (-a^{2}-\lambda \right ) x^{2}-m^{2}+\lambda \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}-a^{2} x^{2}-\lambda \,x^{2}-m^{2}+\lambda \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}-a^{2} x^{2}-\lambda \,x^{2}-m^{2}+\lambda \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}-a^{2} x^{2}-\lambda \,x^{2}-m^{2}+\lambda }{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}-a^{2} x^{2}-\lambda \,x^{2}-m^{2}+\lambda \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}+13 a^{2} u^{4}-12 a^{2} u^{3}-\lambda \,u^{4}+4 a^{2} u^{2}+4 \lambda \,u^{3}-m^{2} u^{2}-4 \lambda \,u^{2}+2 m^{2} u \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 (4 a^{2}-m^{2}+12 r^{2}-4 \lambda +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 (4 a^{2}-m^{2}+12 r^{2}-4 \lambda +28 r +16\right )-2 a_{0} \left (6 a^{2}+3 r^{2}-2 \lambda +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 (4 a^{2}-m^{2}+12 r^{2}-4 \lambda +52 r +56\right )-2 a_{1} \left (6 a^{2}+3 r^{2}-2 \lambda +8 r +5\right )+a_{0} \left (13 a^{2}+r^{2}-\lambda +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 (4 a^{2}-m^{2}+12 r^{2}-4 \lambda +76 r +120\right )-2 a_{2} \left (6 a^{2}+3 r^{2}-2 \lambda +14 r +16\right )+a_{1} \left (13 a^{2}+r^{2}-\lambda +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 (4 a^{2}+12 \left (k -2\right )^{2}+24 \left (k -2\right ) r -m^{2}+12 r^{2}+4 k -8-4 \lambda +4 r \right )-2 a_{k -3} \left (6 a^{2}+3 \left (k -3\right )^{2}+6 \left (k -3\right ) r +3 r^{2}+2 k -6-2 \lambda +2 r \right )+a_{k -4} \left (13 a^{2}+\left (k -4\right )^{2}+2 \left (k -4\right ) r +r^{2}+k -4-\lambda +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 (4 a^{2}-m^{2}+12 r^{2}-4 \lambda +4 r \right )=0, -2 a_{2} \left (4+m +2 r \right ) \left (4-m +2 r \right )+a_{1} \left (4 a^{2}-m^{2}+12 r^{2}-4 \lambda +28 r +16\right )-2 a_{0} \left (6 a^{2}+3 r^{2}-2 \lambda +2 r \right )=0, -2 a_{3} \left (6+m +2 r \right ) \left (6-m +2 r \right )+a_{2} \left (4 a^{2}-m^{2}+12 r^{2}-4 \lambda +52 r +56\right )-2 a_{1} \left (6 a^{2}+3 r^{2}-2 \lambda +8 r +5\right )+a_{0} \left (13 a^{2}+r^{2}-\lambda +r \right )=0, -2 a_{4} \left (8+m +2 r \right ) \left (8-m +2 r \right )+a_{3} \left (4 a^{2}-m^{2}+12 r^{2}-4 \lambda +76 r +120\right )-2 a_{2} \left (6 a^{2}+3 r^{2}-2 \lambda +14 r +16\right )+a_{1} \left (13 a^{2}+r^{2}-\lambda +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 (4 a^{2}-m^{2}+12 r^{2}-4 \lambda +4 r \right )}{2 \left (m^{2}-4 r^{2}-8 r -4\right )}, a_{2}=\frac {a_{0} \left (16 a^{4}+16 a^{2} m^{2}+m^{4}-12 m^{2} r^{2}+96 r^{4}-32 a^{2} \lambda -64 a^{2} r -64 \lambda \,r^{2}-24 m^{2} r +256 r^{3}-32 a^{2}+16 \lambda ^{2}-64 \lambda r -16 m^{2}+192 r^{2}-32 \lambda +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 (64 a^{6}+144 a^{4} m^{2}-192 a^{4} r^{2}+16 a^{2} m^{4}+160 a^{2} m^{2} r^{2}-128 a^{2} r^{4}-m^{6}+16 m^{4} r^{2}-80 m^{2} r^{4}+640 r^{6}-192 a^{4} \lambda -960 a^{4} r -160 a^{2} \lambda \,m^{2}-128 a^{2} \lambda \,r^{2}+160 a^{2} m^{2} r -1024 a^{2} r^{3}-32 \lambda \,m^{2} r^{2}-640 \lambda \,r^{4}+48 m^{4} r -544 m^{2} r^{3}+4480 r^{5}-768 a^{4}+192 a^{2} \lambda ^{2}+384 a^{2} \lambda r +288 a^{2} m^{2}-1408 a^{2} r^{2}+16 \lambda ^{2} m^{2}+320 \lambda ^{2} r^{2}-32 \lambda \,m^{2} r -2560 \lambda \,r^{3}+52 m^{4}-1328 m^{2} r^{2}+11520 r^{4}+256 a^{2} \lambda -64 \lambda ^{3}+576 \lambda ^{2} r -32 \lambda \,m^{2}-3712 \lambda \,r^{2}-1376 m^{2} r +13184 r^{3}+256 a^{2}+512 \lambda ^{2}-2304 \lambda r -576 m^{2}+6272 r^{2}-768 \lambda +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^{8}+896 a^{6} m^{2}-1536 a^{6} r^{2}+512 a^{4} m^{4}-448 a^{4} m^{2} r^{2}-256 a^{4} r^{4}+384 a^{2} m^{4} r^{2}+768 a^{2} m^{2} r^{4}-1024 a^{2} r^{6}+m^{8}-20 r^{2} m^{6}+160 r^{4} m^{4}-320 r^{6} m^{2}+3840 r^{8}-1024 a^{6} \lambda -8192 a^{6} r -1920 a^{4} \lambda \,m^{2}+1536 a^{4} \lambda \,r^{2}-5632 a^{4} m^{2} r +1024 a^{4} r^{3}-256 a^{2} \lambda \,m^{4}-2176 a^{2} \lambda \,m^{2} r^{2}+512 a^{2} \lambda \,r^{4}+1024 a^{2} m^{4} r +1792 a^{2} m^{2} r^{3}-11264 a^{2} r^{5}-768 \lambda \,m^{2} r^{4}-5120 \lambda \,r^{6}-80 r \,m^{6}+1280 r^{3} m^{4}-5376 r^{5} m^{2}+51200 r^{7}-9216 a^{6}+1536 a^{4} \lambda ^{2}+12288 a^{4} \lambda r -5184 a^{4} m^{2}+19712 a^{4} r^{2}+1152 a^{2} \lambda ^{2} m^{2}+1536 a^{2} \lambda ^{2} r^{2}-4096 a^{2} \lambda \,m^{2} r +10240 a^{2} \lambda \,r^{3}+1664 a^{2} m^{4}+5504 a^{2} m^{2} r^{2}-36352 a^{2} r^{4}+576 \lambda ^{2} m^{2} r^{2}+3840 \lambda ^{2} r^{4}-3840 \lambda \,m^{2} r^{3}-46080 \lambda \,r^{5}-116 m^{6}+4080 m^{4} r^{2}-32640 m^{2} r^{4}+276480 r^{6}+13312 a^{4} \lambda +43008 a^{4} r -1024 a^{2} \lambda ^{3}-7040 a^{2} \lambda \,m^{2}+22016 a^{2} \lambda \,r^{2}+12800 a^{2} m^{2} r -36864 a^{2} r^{3}-128 \lambda ^{3} m^{2}-1536 \lambda ^{3} r^{2}+1536 \lambda ^{2} m^{2} r +21504 \lambda ^{2} r^{3}-7296 \lambda \,m^{2} r^{2}-163328 \lambda \,r^{4}+6080 m^{4} r -95360 m^{2} r^{3}+774656 r^{5}+19456 a^{4}+1024 a^{2} \lambda ^{2}+4096 a^{2} \lambda r +12928 a^{2} m^{2}+7680 a^{2} r^{2}+256 \lambda ^{4}-4096 \lambda ^{3} r +1984 \lambda ^{2} m^{2}+48384 \lambda ^{2} r^{2}-6144 \lambda \,m^{2} r -288768 \lambda \,r^{3}+3904 m^{4}-144960 m^{2} r^{2}+1194240 r^{4}-10240 a^{2} \lambda +28672 a^{2} r -5120 \lambda ^{3}+51200 \lambda ^{2} r -3456 \lambda \,m^{2}-270848 \lambda \,r^{2}-111744 m^{2} r +980480 r^{3}+12288 a^{2}+27648 \lambda ^{2}-135168 \lambda r -36864 m^{2}+368640 r^{2}-36864 \lambda +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 (4 a^{2}-m^{2}-4 \lambda +40\right ) a_{k -2}+\left (13 a^{2}-\lambda +12\right ) a_{k -4}+2 \left (-6 a^{2}+2 \lambda -21\right ) a_{k -3}+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 (4 a^{2}-m^{2}-4 \lambda +40\right ) a_{k +4}+\left (13 a^{2}-\lambda +12\right ) a_{k +2}+2 \left (-6 a^{2}+2 \lambda -21\right ) a_{k +3}+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^{2} a_{k}-6 a^{2} a_{k +1}+13 a^{2} a_{k +2}-12 a^{2} a_{k +3}+4 a^{2} a_{k +4}+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}+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}-\lambda a_{k +2}+4 \lambda a_{k +3}-4 \lambda 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}+\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}+a^{2} a_{k}-6 a^{2} a_{k +1}+13 a^{2} a_{k +2}-12 a^{2} a_{k +3}+4 a^{2} a_{k +4}+k^{2} a_{k +2}-6 k^{2} a_{k +3}+12 k^{2} a_{k +4}+2 m^{2} a_{k +4}+5 k a_{k +2}-40 k a_{k +3}+100 k a_{k +4}-\lambda a_{k +2}+4 \lambda a_{k +3}-4 \lambda a_{k +4}+208 a_{k +4}+6 a_{k +2}-66 a_{k +3}}{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}+\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}+a^{2} a_{k}-6 a^{2} a_{k +1}+13 a^{2} a_{k +2}-12 a^{2} a_{k +3}+4 a^{2} a_{k +4}+k^{2} a_{k +2}-6 k^{2} a_{k +3}+12 k^{2} a_{k +4}+2 m^{2} a_{k +4}+5 k a_{k +2}-40 k a_{k +3}+100 k a_{k +4}-\lambda a_{k +2}+4 \lambda a_{k +3}-4 \lambda a_{k +4}+208 a_{k +4}+6 a_{k +2}-66 a_{k +3}}{2 \left (4 k^{2}-4 k m +40 k -20 m +100\right )}, a_{1}=-\frac {a_{0} \left (4 a^{2}+2 m^{2}-4 \lambda -2 m \right )}{2 \left (4 m -4\right )}, a_{2}=\frac {a_{0} \left (16 a^{4}+16 a^{2} m^{2}+4 m^{4}-32 a^{2} \lambda +32 a^{2} m -16 \lambda \,m^{2}-20 m^{3}-32 a^{2}+16 \lambda ^{2}+32 \lambda m +32 m^{2}-32 \lambda -16 m \right )}{4 \left (32 m^{2}-96 m +64\right )}, a_{3}=-\frac {a_{0} \left (64 a^{6}+96 a^{4} m^{2}+48 a^{2} m^{4}+8 m^{6}-192 a^{4} \lambda +480 a^{4} m -192 a^{2} \lambda \,m^{2}+48 a^{2} m^{3}-48 \lambda \,m^{4}-96 m^{5}-768 a^{4}+192 a^{2} \lambda ^{2}-192 a^{2} \lambda m -64 a^{2} m^{2}+96 \lambda ^{2} m^{2}+336 \lambda \,m^{3}+440 m^{4}+256 a^{2} \lambda -64 \lambda ^{3}-288 \lambda ^{2} m -960 \lambda \,m^{2}-960 m^{3}+256 a^{2}+512 \lambda ^{2}+1152 \lambda m +992 m^{2}-768 \lambda -384 m \right )}{8 \left (384 m^{3}-2304 m^{2}+4224 m -2304\right )}, a_{4}=\frac {a_{0} \left (256 a^{8}+512 a^{6} m^{2}+384 a^{4} m^{4}+128 a^{2} m^{6}+16 m^{8}-1024 a^{6} \lambda +4096 a^{6} m -1536 a^{4} \lambda \,m^{2}+2688 a^{4} m^{3}-768 a^{2} \lambda \,m^{4}-384 a^{2} m^{5}-128 \lambda \,m^{6}-352 m^{7}-9216 a^{6}+1536 a^{4} \lambda ^{2}-6144 a^{4} \lambda m -256 a^{4} m^{2}+1536 a^{2} \lambda ^{2} m^{2}+768 a^{2} \lambda \,m^{3}+768 a^{2} m^{4}+384 \lambda ^{2} m^{4}+1920 \lambda \,m^{5}+3184 m^{6}+13312 a^{4} \lambda -21504 a^{4} m -1024 a^{2} \lambda ^{3}-1536 a^{2} \lambda \,m^{2}-1792 a^{2} m^{3}-512 \lambda ^{3} m^{2}-3456 \lambda ^{2} m^{3}-12032 \lambda \,m^{4}-15328 m^{5}+19456 a^{4}+1024 a^{2} \lambda ^{2}-2048 a^{2} \lambda m +14848 a^{2} m^{2}+256 \lambda ^{4}+2048 \lambda ^{3} m +14080 \lambda ^{2} m^{2}+39168 \lambda \,m^{3}+42304 m^{4}-10240 a^{2} \lambda -14336 a^{2} m -5120 \lambda ^{3}-25600 \lambda ^{2} m -71168 \lambda \,m^{2}-66688 m^{3}+12288 a^{2}+27648 \lambda ^{2}+67584 \lambda m +55296 m^{2}-36864 \lambda -18432 m \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}+\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}+a^{2} a_{k}-6 a^{2} a_{k +1}+13 a^{2} a_{k +2}-12 a^{2} a_{k +3}+4 a^{2} a_{k +4}+k^{2} a_{k +2}-6 k^{2} a_{k +3}+12 k^{2} a_{k +4}+2 m^{2} a_{k +4}+5 k a_{k +2}-40 k a_{k +3}+100 k a_{k +4}-\lambda a_{k +2}+4 \lambda a_{k +3}-4 \lambda a_{k +4}+208 a_{k +4}+6 a_{k +2}-66 a_{k +3}}{2 \left (4 k^{2}-4 k m +40 k -20 m +100\right )}, a_{1}=-\frac {a_{0} \left (4 a^{2}+2 m^{2}-4 \lambda -2 m \right )}{2 \left (4 m -4\right )}, a_{2}=\frac {a_{0} \left (16 a^{4}+16 a^{2} m^{2}+4 m^{4}-32 a^{2} \lambda +32 a^{2} m -16 \lambda \,m^{2}-20 m^{3}-32 a^{2}+16 \lambda ^{2}+32 \lambda m +32 m^{2}-32 \lambda -16 m \right )}{4 \left (32 m^{2}-96 m +64\right )}, a_{3}=-\frac {a_{0} \left (64 a^{6}+96 a^{4} m^{2}+48 a^{2} m^{4}+8 m^{6}-192 a^{4} \lambda +480 a^{4} m -192 a^{2} \lambda \,m^{2}+48 a^{2} m^{3}-48 \lambda \,m^{4}-96 m^{5}-768 a^{4}+192 a^{2} \lambda ^{2}-192 a^{2} \lambda m -64 a^{2} m^{2}+96 \lambda ^{2} m^{2}+336 \lambda \,m^{3}+440 m^{4}+256 a^{2} \lambda -64 \lambda ^{3}-288 \lambda ^{2} m -960 \lambda \,m^{2}-960 m^{3}+256 a^{2}+512 \lambda ^{2}+1152 \lambda m +992 m^{2}-768 \lambda -384 m \right )}{8 \left (384 m^{3}-2304 m^{2}+4224 m -2304\right )}, a_{4}=\frac {a_{0} \left (256 a^{8}+512 a^{6} m^{2}+384 a^{4} m^{4}+128 a^{2} m^{6}+16 m^{8}-1024 a^{6} \lambda +4096 a^{6} m -1536 a^{4} \lambda \,m^{2}+2688 a^{4} m^{3}-768 a^{2} \lambda \,m^{4}-384 a^{2} m^{5}-128 \lambda \,m^{6}-352 m^{7}-9216 a^{6}+1536 a^{4} \lambda ^{2}-6144 a^{4} \lambda m -256 a^{4} m^{2}+1536 a^{2} \lambda ^{2} m^{2}+768 a^{2} \lambda \,m^{3}+768 a^{2} m^{4}+384 \lambda ^{2} m^{4}+1920 \lambda \,m^{5}+3184 m^{6}+13312 a^{4} \lambda -21504 a^{4} m -1024 a^{2} \lambda ^{3}-1536 a^{2} \lambda \,m^{2}-1792 a^{2} m^{3}-512 \lambda ^{3} m^{2}-3456 \lambda ^{2} m^{3}-12032 \lambda \,m^{4}-15328 m^{5}+19456 a^{4}+1024 a^{2} \lambda ^{2}-2048 a^{2} \lambda m +14848 a^{2} m^{2}+256 \lambda ^{4}+2048 \lambda ^{3} m +14080 \lambda ^{2} m^{2}+39168 \lambda \,m^{3}+42304 m^{4}-10240 a^{2} \lambda -14336 a^{2} m -5120 \lambda ^{3}-25600 \lambda ^{2} m -71168 \lambda \,m^{2}-66688 m^{3}+12288 a^{2}+27648 \lambda ^{2}+67584 \lambda m +55296 m^{2}-36864 \lambda -18432 m \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}+\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}+a^{2} a_{k}-6 a^{2} a_{k +1}+13 a^{2} a_{k +2}-12 a^{2} a_{k +3}+4 a^{2} a_{k +4}+k^{2} a_{k +2}-6 k^{2} a_{k +3}+12 k^{2} a_{k +4}+2 m^{2} a_{k +4}+5 k a_{k +2}-40 k a_{k +3}+100 k a_{k +4}-\lambda a_{k +2}+4 \lambda a_{k +3}-4 \lambda a_{k +4}+208 a_{k +4}+6 a_{k +2}-66 a_{k +3}}{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}+\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}+a^{2} a_{k}-6 a^{2} a_{k +1}+13 a^{2} a_{k +2}-12 a^{2} a_{k +3}+4 a^{2} a_{k +4}+k^{2} a_{k +2}-6 k^{2} a_{k +3}+12 k^{2} a_{k +4}+2 m^{2} a_{k +4}+5 k a_{k +2}-40 k a_{k +3}+100 k a_{k +4}-\lambda a_{k +2}+4 \lambda a_{k +3}-4 \lambda a_{k +4}+208 a_{k +4}+6 a_{k +2}-66 a_{k +3}}{2 \left (4 k^{2}+4 k m +40 k +20 m +100\right )}, a_{1}=-\frac {a_{0} \left (4 a^{2}+2 m^{2}-4 \lambda +2 m \right )}{2 \left (-4 m -4\right )}, a_{2}=\frac {a_{0} \left (16 a^{4}+16 a^{2} m^{2}+4 m^{4}-32 a^{2} \lambda -32 a^{2} m -16 \lambda \,m^{2}+20 m^{3}-32 a^{2}+16 \lambda ^{2}-32 \lambda m +32 m^{2}-32 \lambda +16 m \right )}{4 \left (32 m^{2}+96 m +64\right )}, a_{3}=-\frac {a_{0} \left (64 a^{6}+96 a^{4} m^{2}+48 a^{2} m^{4}+8 m^{6}-192 a^{4} \lambda -480 a^{4} m -192 a^{2} \lambda \,m^{2}-48 a^{2} m^{3}-48 \lambda \,m^{4}+96 m^{5}-768 a^{4}+192 a^{2} \lambda ^{2}+192 a^{2} \lambda m -64 a^{2} m^{2}+96 \lambda ^{2} m^{2}-336 \lambda \,m^{3}+440 m^{4}+256 a^{2} \lambda -64 \lambda ^{3}+288 \lambda ^{2} m -960 \lambda \,m^{2}+960 m^{3}+256 a^{2}+512 \lambda ^{2}-1152 \lambda m +992 m^{2}-768 \lambda +384 m \right )}{8 \left (-384 m^{3}-2304 m^{2}-4224 m -2304\right )}, a_{4}=\frac {a_{0} \left (256 a^{8}+512 a^{6} m^{2}+384 a^{4} m^{4}+128 a^{2} m^{6}+16 m^{8}-1024 a^{6} \lambda -4096 a^{6} m -1536 a^{4} \lambda \,m^{2}-2688 a^{4} m^{3}-768 a^{2} \lambda \,m^{4}+384 a^{2} m^{5}-128 \lambda \,m^{6}+352 m^{7}-9216 a^{6}+1536 a^{4} \lambda ^{2}+6144 a^{4} \lambda m -256 a^{4} m^{2}+1536 a^{2} \lambda ^{2} m^{2}-768 a^{2} \lambda \,m^{3}+768 a^{2} m^{4}+384 \lambda ^{2} m^{4}-1920 \lambda \,m^{5}+3184 m^{6}+13312 a^{4} \lambda +21504 a^{4} m -1024 a^{2} \lambda ^{3}-1536 a^{2} \lambda \,m^{2}+1792 a^{2} m^{3}-512 \lambda ^{3} m^{2}+3456 \lambda ^{2} m^{3}-12032 \lambda \,m^{4}+15328 m^{5}+19456 a^{4}+1024 a^{2} \lambda ^{2}+2048 a^{2} \lambda m +14848 a^{2} m^{2}+256 \lambda ^{4}-2048 \lambda ^{3} m +14080 \lambda ^{2} m^{2}-39168 \lambda \,m^{3}+42304 m^{4}-10240 a^{2} \lambda +14336 a^{2} m -5120 \lambda ^{3}+25600 \lambda ^{2} m -71168 \lambda \,m^{2}+66688 m^{3}+12288 a^{2}+27648 \lambda ^{2}-67584 \lambda m +55296 m^{2}-36864 \lambda +18432 m \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}+\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}+a^{2} a_{k}-6 a^{2} a_{k +1}+13 a^{2} a_{k +2}-12 a^{2} a_{k +3}+4 a^{2} a_{k +4}+k^{2} a_{k +2}-6 k^{2} a_{k +3}+12 k^{2} a_{k +4}+2 m^{2} a_{k +4}+5 k a_{k +2}-40 k a_{k +3}+100 k a_{k +4}-\lambda a_{k +2}+4 \lambda a_{k +3}-4 \lambda a_{k +4}+208 a_{k +4}+6 a_{k +2}-66 a_{k +3}}{2 \left (4 k^{2}+4 k m +40 k +20 m +100\right )}, a_{1}=-\frac {a_{0} \left (4 a^{2}+2 m^{2}-4 \lambda +2 m \right )}{2 \left (-4 m -4\right )}, a_{2}=\frac {a_{0} \left (16 a^{4}+16 a^{2} m^{2}+4 m^{4}-32 a^{2} \lambda -32 a^{2} m -16 \lambda \,m^{2}+20 m^{3}-32 a^{2}+16 \lambda ^{2}-32 \lambda m +32 m^{2}-32 \lambda +16 m \right )}{4 \left (32 m^{2}+96 m +64\right )}, a_{3}=-\frac {a_{0} \left (64 a^{6}+96 a^{4} m^{2}+48 a^{2} m^{4}+8 m^{6}-192 a^{4} \lambda -480 a^{4} m -192 a^{2} \lambda \,m^{2}-48 a^{2} m^{3}-48 \lambda \,m^{4}+96 m^{5}-768 a^{4}+192 a^{2} \lambda ^{2}+192 a^{2} \lambda m -64 a^{2} m^{2}+96 \lambda ^{2} m^{2}-336 \lambda \,m^{3}+440 m^{4}+256 a^{2} \lambda -64 \lambda ^{3}+288 \lambda ^{2} m -960 \lambda \,m^{2}+960 m^{3}+256 a^{2}+512 \lambda ^{2}-1152 \lambda m +992 m^{2}-768 \lambda +384 m \right )}{8 \left (-384 m^{3}-2304 m^{2}-4224 m -2304\right )}, a_{4}=\frac {a_{0} \left (256 a^{8}+512 a^{6} m^{2}+384 a^{4} m^{4}+128 a^{2} m^{6}+16 m^{8}-1024 a^{6} \lambda -4096 a^{6} m -1536 a^{4} \lambda \,m^{2}-2688 a^{4} m^{3}-768 a^{2} \lambda \,m^{4}+384 a^{2} m^{5}-128 \lambda \,m^{6}+352 m^{7}-9216 a^{6}+1536 a^{4} \lambda ^{2}+6144 a^{4} \lambda m -256 a^{4} m^{2}+1536 a^{2} \lambda ^{2} m^{2}-768 a^{2} \lambda \,m^{3}+768 a^{2} m^{4}+384 \lambda ^{2} m^{4}-1920 \lambda \,m^{5}+3184 m^{6}+13312 a^{4} \lambda +21504 a^{4} m -1024 a^{2} \lambda ^{3}-1536 a^{2} \lambda \,m^{2}+1792 a^{2} m^{3}-512 \lambda ^{3} m^{2}+3456 \lambda ^{2} m^{3}-12032 \lambda \,m^{4}+15328 m^{5}+19456 a^{4}+1024 a^{2} \lambda ^{2}+2048 a^{2} \lambda m +14848 a^{2} m^{2}+256 \lambda ^{4}-2048 \lambda ^{3} m +14080 \lambda ^{2} m^{2}-39168 \lambda \,m^{3}+42304 m^{4}-10240 a^{2} \lambda +14336 a^{2} m -5120 \lambda ^{3}+25600 \lambda ^{2} m -71168 \lambda \,m^{2}+66688 m^{3}+12288 a^{2}+27648 \lambda ^{2}-67584 \lambda m +55296 m^{2}-36864 \lambda +18432 m \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 {-k m b_{k +2}+6 k m b_{k +3}-12 k m 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}+a^{2} b_{k}-6 a^{2} b_{k +1}+13 a^{2} b_{k +2}-12 a^{2} b_{k +3}+4 a^{2} b_{k +4}+k^{2} b_{k +2}-6 k^{2} b_{k +3}+12 k^{2} b_{k +4}+2 m^{2} b_{k +4}+5 k b_{k +2}-40 k b_{k +3}+100 k b_{k +4}-\lambda b_{k +2}+4 \lambda b_{k +3}-4 \lambda b_{k +4}+6 b_{k +2}-66 b_{k +3}+208 b_{k +4}}{2 \left (4 k^{2}-4 k m +40 k -20 m +100\right )}, b_{1}=-\frac {b_{0} \left (4 a^{2}+2 m^{2}-4 \lambda -2 m \right )}{2 \left (4 m -4\right )}, b_{2}=\frac {b_{0} \left (16 a^{4}+16 a^{2} m^{2}+4 m^{4}-32 a^{2} \lambda +32 a^{2} m -16 \lambda \,m^{2}-20 m^{3}-32 a^{2}+16 \lambda ^{2}+32 \lambda m +32 m^{2}-32 \lambda -16 m \right )}{4 \left (32 m^{2}-96 m +64\right )}, b_{3}=-\frac {b_{0} \left (64 a^{6}+96 a^{4} m^{2}+48 a^{2} m^{4}+8 m^{6}-192 a^{4} \lambda +480 a^{4} m -192 a^{2} \lambda \,m^{2}+48 a^{2} m^{3}-48 \lambda \,m^{4}-96 m^{5}-768 a^{4}+192 a^{2} \lambda ^{2}-192 a^{2} \lambda m -64 a^{2} m^{2}+96 \lambda ^{2} m^{2}+336 \lambda \,m^{3}+440 m^{4}+256 a^{2} \lambda -64 \lambda ^{3}-288 \lambda ^{2} m -960 \lambda \,m^{2}-960 m^{3}+256 a^{2}+512 \lambda ^{2}+1152 \lambda m +992 m^{2}-768 \lambda -384 m \right )}{8 \left (384 m^{3}-2304 m^{2}+4224 m -2304\right )}, b_{4}=\frac {b_{0} \left (256 a^{8}+512 a^{6} m^{2}+384 a^{4} m^{4}+128 a^{2} m^{6}+16 m^{8}-1024 a^{6} \lambda +4096 a^{6} m -1536 a^{4} \lambda \,m^{2}+2688 a^{4} m^{3}-768 a^{2} \lambda \,m^{4}-384 a^{2} m^{5}-128 \lambda \,m^{6}-352 m^{7}-9216 a^{6}+1536 a^{4} \lambda ^{2}-6144 a^{4} \lambda m -256 a^{4} m^{2}+1536 a^{2} \lambda ^{2} m^{2}+768 a^{2} \lambda \,m^{3}+768 a^{2} m^{4}+384 \lambda ^{2} m^{4}+1920 \lambda \,m^{5}+3184 m^{6}+13312 a^{4} \lambda -21504 a^{4} m -1024 a^{2} \lambda ^{3}-1536 a^{2} \lambda \,m^{2}-1792 a^{2} m^{3}-512 \lambda ^{3} m^{2}-3456 \lambda ^{2} m^{3}-12032 \lambda \,m^{4}-15328 m^{5}+19456 a^{4}+1024 a^{2} \lambda ^{2}-2048 a^{2} \lambda m +14848 a^{2} m^{2}+256 \lambda ^{4}+2048 \lambda ^{3} m +14080 \lambda ^{2} m^{2}+39168 \lambda \,m^{3}+42304 m^{4}-10240 a^{2} \lambda -14336 a^{2} m -5120 \lambda ^{3}-25600 \lambda ^{2} m -71168 \lambda \,m^{2}-66688 m^{3}+12288 a^{2}+27648 \lambda ^{2}+67584 \lambda m +55296 m^{2}-36864 \lambda -18432 m \right )}{16 \left (6144 m^{4}-61440 m^{3}+215040 m^{2}-307200 m +147456\right )}, c_{k +5}=\frac {k m c_{k +2}-6 k m c_{k +3}+12 k m 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}+a^{2} c_{k}-6 a^{2} c_{k +1}+13 a^{2} c_{k +2}-12 a^{2} c_{k +3}+4 a^{2} c_{k +4}+k^{2} c_{k +2}-6 k^{2} c_{k +3}+12 k^{2} c_{k +4}+2 m^{2} c_{k +4}+5 k c_{k +2}-40 k c_{k +3}+100 k c_{k +4}-\lambda c_{k +2}+4 \lambda c_{k +3}-4 \lambda c_{k +4}-66 c_{k +3}+208 c_{k +4}+6 c_{k +2}}{2 \left (4 k^{2}+4 k m +40 k +20 m +100\right )}, c_{1}=-\frac {c_{0} \left (4 a^{2}+2 m^{2}-4 \lambda +2 m \right )}{2 \left (-4 m -4\right )}, c_{2}=\frac {c_{0} \left (16 a^{4}+16 a^{2} m^{2}+4 m^{4}-32 a^{2} \lambda -32 a^{2} m -16 \lambda \,m^{2}+20 m^{3}-32 a^{2}+16 \lambda ^{2}-32 \lambda m +32 m^{2}-32 \lambda +16 m \right )}{4 \left (32 m^{2}+96 m +64\right )}, c_{3}=-\frac {c_{0} \left (64 a^{6}+96 a^{4} m^{2}+48 a^{2} m^{4}+8 m^{6}-192 a^{4} \lambda -480 a^{4} m -192 a^{2} \lambda \,m^{2}-48 a^{2} m^{3}-48 \lambda \,m^{4}+96 m^{5}-768 a^{4}+192 a^{2} \lambda ^{2}+192 a^{2} \lambda m -64 a^{2} m^{2}+96 \lambda ^{2} m^{2}-336 \lambda \,m^{3}+440 m^{4}+256 a^{2} \lambda -64 \lambda ^{3}+288 \lambda ^{2} m -960 \lambda \,m^{2}+960 m^{3}+256 a^{2}+512 \lambda ^{2}-1152 \lambda m +992 m^{2}-768 \lambda +384 m \right )}{8 \left (-384 m^{3}-2304 m^{2}-4224 m -2304\right )}, c_{4}=\frac {c_{0} \left (256 a^{8}+512 a^{6} m^{2}+384 a^{4} m^{4}+128 a^{2} m^{6}+16 m^{8}-1024 a^{6} \lambda -4096 a^{6} m -1536 a^{4} \lambda \,m^{2}-2688 a^{4} m^{3}-768 a^{2} \lambda \,m^{4}+384 a^{2} m^{5}-128 \lambda \,m^{6}+352 m^{7}-9216 a^{6}+1536 a^{4} \lambda ^{2}+6144 a^{4} \lambda m -256 a^{4} m^{2}+1536 a^{2} \lambda ^{2} m^{2}-768 a^{2} \lambda \,m^{3}+768 a^{2} m^{4}+384 \lambda ^{2} m^{4}-1920 \lambda \,m^{5}+3184 m^{6}+13312 a^{4} \lambda +21504 a^{4} m -1024 a^{2} \lambda ^{3}-1536 a^{2} \lambda \,m^{2}+1792 a^{2} m^{3}-512 \lambda ^{3} m^{2}+3456 \lambda ^{2} m^{3}-12032 \lambda \,m^{4}+15328 m^{5}+19456 a^{4}+1024 a^{2} \lambda ^{2}+2048 a^{2} \lambda m +14848 a^{2} m^{2}+256 \lambda ^{4}-2048 \lambda ^{3} m +14080 \lambda ^{2} m^{2}-39168 \lambda \,m^{3}+42304 m^{4}-10240 a^{2} \lambda +14336 a^{2} m -5120 \lambda ^{3}+25600 \lambda ^{2} m -71168 \lambda \,m^{2}+66688 m^{3}+12288 a^{2}+27648 \lambda ^{2}-67584 \lambda m +55296 m^{2}-36864 \lambda +18432 m \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: 0.5 (sec). Leaf size: 64
dsolve((x^2-1)^2*diff(y(x),x$2)+2*x*(x^2-1)*diff(y(x),x)+( (x^2-1)*(a^2*x^2-lambda)-m^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\operatorname {HeunC}\left (0, \frac {1}{2}, m , \frac {a^{2}}{4}, \frac {1}{4}+\frac {m^{2}}{4}-\frac {\lambda }{4}, x^{2}\right ) c_{2} x +\operatorname {HeunC}\left (0, -\frac {1}{2}, m , \frac {a^{2}}{4}, \frac {1}{4}+\frac {m^{2}}{4}-\frac {\lambda }{4}, x^{2}\right ) c_{1} \right ) \left (x^{2}-1\right )^{\frac {m}{2}} \]
✓ Solution by Mathematica
Time used: 0.602 (sec). Leaf size: 234
DSolve[(x^2-1)^2*y''[x]+2*x*(x^2-1)*y'[x]+( (x^2-1)*(a^2*x^2-\[Lambda])-m^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{i \sqrt {a^2} x} \left (\frac {x+1}{x-1}\right )^{\frac {\sqrt {m^2}}{2}} \left (c_2 (x-1)^{\sqrt {m^2}} \text {HeunC}\left [-\left (\sqrt {m^2}+1\right ) \left (\sqrt {m^2}+2 i \sqrt {a^2}\right )-a^2+\lambda ,-4 i \sqrt {a^2} \left (\sqrt {m^2}+1\right ),\sqrt {m^2}+1,\sqrt {m^2}+1,-4 i \sqrt {a^2},\frac {1-x}{2}\right ]+c_1 \text {HeunC}\left [2 i \sqrt {a^2} \left (\sqrt {m^2}-1\right )-a^2+\lambda ,-4 i \sqrt {a^2},1-\sqrt {m^2},\sqrt {m^2}+1,-4 i \sqrt {a^2},\frac {1-x}{2}\right ]\right ) \]