2.33.26 Problem 235

Internal problem ID [13896]
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
Problem number : 235
Date solved : Friday, December 19, 2025 at 08:06:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \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 \\ \end{align*}

2.33.26.1 ✓ Maple. Time used: 0.048 (sec). Leaf size: 64

ode:=(x^2-1)^2*diff(diff(y(x),x),x)+2*x*(x^2-1)*diff(y(x),x)+((x^2-1)*(a^2*x^2-lambda)-m^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\operatorname {HeunC}\left (0, \frac {1}{2}, m , \frac {a^{2}}{4}, \frac {1}{4}+\frac {m^{2}}{4}-\frac {\lambda }{4}, x^{2}\right ) c_2 x +\operatorname {HeunC}\left (0, -\frac {1}{2}, m , \frac {a^{2}}{4}, \frac {1}{4}+\frac {m^{2}}{4}-\frac {\lambda }{4}, x^{2}\right ) c_1 \right ) \left (x^{2}-1\right )^{\frac {m}{2}} \]

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 @ Moebi\ 
us 
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
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{2}-1\right )^{2} \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+2 x \left (x^{2}-1\right ) \left (\frac {d}{d x}y \left (x \right )\right )+\left (\left (x^{2}-1\right ) \left (a^{2} x^{2}-\lambda \right )-m^{2}\right ) y \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=-\frac {\left (a^{2} x^{4}-a^{2} x^{2}-\lambda \,x^{2}-m^{2}+\lambda \right ) y \left (x \right )}{\left (x^{2}-1\right )^{2}}-\frac {2 x \left (\frac {d}{d x}y \left (x \right )\right )}{x^{2}-1} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y \left (x \right )\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^{2}}{d x^{2}}y \left (x \right )+\frac {2 x \left (\frac {d}{d x}y \left (x \right )\right )}{x^{2}-1}+\frac {\left (a^{2} x^{4}-a^{2} x^{2}-\lambda \,x^{2}-m^{2}+\lambda \right ) y \left (x \right )}{\left (x^{2}-1\right )^{2}}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {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 }{\left (x^{2}-1\right )^{2}}\right ] \\ {} & \circ & \left (x +1\right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-1 \\ {} & {} & \left (\left (x +1\right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-1}}}=1 \\ {} & \circ & \left (x +1\right )^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-1 \\ {} & {} & \left (\left (x +1\right )^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-1}}}=-\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 (x^{2}-1\right )^{2} \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+2 x \left (x^{2}-1\right ) \left (\frac {d}{d x}y \left (x \right )\right )+\left (a^{2} x^{4}-a^{2} x^{2}-\lambda \,x^{2}-m^{2}+\lambda \right ) y \left (x \right )=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^{4}-4 u^{3}+4 u^{2}\right ) \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (2 u^{3}-6 u^{2}+4 u \right ) \left (\frac {d}{d u}y \left (u \right )\right )+\left (a^{2} u^{4}-4 a^{2} u^{3}+5 a^{2} u^{2}-2 a^{2} u -\lambda \,u^{2}+2 \lambda u -m^{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 =0..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 =1..3 \\ {} & {} & 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^{2}}{d u^{2}}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =2..4 \\ {} & {} & u^{m}\cdot \left (\frac {d^{2}}{d u^{2}}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^{2}}{d u^{2}}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-2+m}{\sum }}a_{k +2-m} \left (k +2-m +r \right ) \left (k +1-m +r \right ) u^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (m +2 r \right ) \left (-m +2 r \right ) u^{r}+\left (a_{1} \left (2+m +2 r \right ) \left (2-m +2 r \right )-2 a_{0} \left (a^{2}+2 r^{2}-\lambda +r \right )\right ) u^{1+r}+\left (a_{2} \left (4+m +2 r \right ) \left (4-m +2 r \right )-2 a_{1} \left (a^{2}+2 r^{2}-\lambda +5 r +3\right )+a_{0} \left (5 a^{2}+r^{2}-\lambda +r \right )\right ) u^{2+r}+\left (a_{3} \left (6+m +2 r \right ) \left (6-m +2 r \right )-2 a_{2} \left (a^{2}+2 r^{2}-\lambda +9 r +10\right )+a_{1} \left (5 a^{2}+r^{2}-\lambda +3 r +2\right )-4 a_{0} a^{2}\right ) u^{3+r}+\left (\moverset {\infty }{\munderset {k =4}{\sum }}\left (a_{k} \left (2 k +m +2 r \right ) \left (2 k -m +2 r \right )-2 a_{k -1} \left (a^{2}+2 \left (k -1\right )^{2}+4 \left (k -1\right ) r +2 r^{2}+k -1-\lambda +r \right )+a_{k -2} \left (5 a^{2}+\left (k -2\right )^{2}+2 \left (k -2\right ) r +r^{2}+k -2-\lambda +r \right )-4 a_{k -3} a^{2}+a_{k -4} a^{2}\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & \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 [a_{1} \left (2+m +2 r \right ) \left (2-m +2 r \right )-2 a_{0} \left (a^{2}+2 r^{2}-\lambda +r \right )=0, a_{2} \left (4+m +2 r \right ) \left (4-m +2 r \right )-2 a_{1} \left (a^{2}+2 r^{2}-\lambda +5 r +3\right )+a_{0} \left (5 a^{2}+r^{2}-\lambda +r \right )=0, a_{3} \left (6+m +2 r \right ) \left (6-m +2 r \right )-2 a_{2} \left (a^{2}+2 r^{2}-\lambda +9 r +10\right )+a_{1} \left (5 a^{2}+r^{2}-\lambda +3 r +2\right )-4 a_{0} a^{2}=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{1}=-\frac {2 a_{0} \left (a^{2}+2 r^{2}-\lambda +r \right )}{m^{2}-4 r^{2}-8 r -4}, a_{2}=\frac {a_{0} \left (4 a^{4}+5 a^{2} m^{2}-4 a^{2} r^{2}+m^{2} r^{2}+12 r^{4}-8 a^{2} \lambda -16 a^{2} r -\lambda \,m^{2}-12 \lambda \,r^{2}+m^{2} r +36 r^{3}-8 a^{2}+4 \lambda ^{2}-16 \lambda r +32 r^{2}-8 \lambda +8 r \right )}{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}, a_{3}=-\frac {4 a_{0} \left (2 a^{6}+5 a^{4} m^{2}-8 a^{4} r^{2}+a^{2} m^{4}+3 a^{2} m^{2} r^{2}-4 a^{2} r^{4}+2 m^{2} r^{4}+8 r^{6}-6 a^{4} \lambda -30 a^{4} r -6 a^{2} \lambda \,m^{2}+3 a^{2} m^{2} r -24 a^{2} r^{3}-3 \lambda \,m^{2} r^{2}-12 \lambda \,r^{4}+9 m^{2} r^{3}+60 r^{5}-24 a^{4}+6 a^{2} \lambda ^{2}+12 a^{2} \lambda r +6 a^{2} m^{2}-32 a^{2} r^{2}+\lambda ^{2} m^{2}+8 \lambda ^{2} r^{2}-7 \lambda \,m^{2} r -56 \lambda \,r^{3}+13 m^{2} r^{2}+168 r^{4}+8 a^{2} \lambda -2 \lambda ^{3}+18 \lambda ^{2} r -6 \lambda \,m^{2}-96 \lambda \,r^{2}+6 m^{2} r +216 r^{3}+8 a^{2}+16 \lambda ^{2}-72 \lambda r +124 r^{2}-24 \lambda +24 r \right )}{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 \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k^{2}+\left (2 r -3\right ) k +5 a^{2}+r^{2}-3 r -\lambda +2\right ) a_{k -2}+\left (-4 k^{2}+\left (-8 r +6\right ) k -2 a^{2}-4 r^{2}+6 r +2 \lambda -2\right ) a_{k -1}+4 k^{2} a_{k}+8 k r a_{k}+4 r^{2} a_{k}-a_{k} m^{2}+a^{2} \left (a_{k -4}-4 a_{k -3}\right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +4 \\ {} & {} & \left (\left (k +4\right )^{2}+\left (2 r -3\right ) \left (k +4\right )+5 a^{2}+r^{2}-3 r -\lambda +2\right ) a_{k +2}+\left (-4 \left (k +4\right )^{2}+\left (-8 r +6\right ) \left (k +4\right )-2 a^{2}-4 r^{2}+6 r +2 \lambda -2\right ) a_{k +3}+4 \left (k +4\right )^{2} a_{k +4}+8 \left (k +4\right ) r a_{k +4}+4 r^{2} a_{k +4}-a_{k +4} m^{2}+a^{2} \left (a_{k}-4 a_{k +1}\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +4}=-\frac {a_{k} a^{2}-4 a^{2} a_{k +1}+5 a^{2} a_{k +2}-2 a^{2} a_{k +3}+k^{2} a_{k +2}-4 k^{2} a_{k +3}+2 k r a_{k +2}-8 k r a_{k +3}+r^{2} a_{k +2}-4 r^{2} a_{k +3}+5 k a_{k +2}-26 k a_{k +3}-\lambda a_{k +2}+2 \lambda a_{k +3}+5 r a_{k +2}-26 r a_{k +3}+6 a_{k +2}-42 a_{k +3}}{4 k^{2}+8 k r -m^{2}+4 r^{2}+32 k +32 r +64} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {m}{2} \\ {} & {} & a_{k +4}=-\frac {a_{k} a^{2}-4 a^{2} a_{k +1}+5 a^{2} a_{k +2}-2 a^{2} a_{k +3}+k^{2} a_{k +2}-4 k^{2} a_{k +3}-k m a_{k +2}+4 k m a_{k +3}+\frac {1}{4} m^{2} a_{k +2}-m^{2} a_{k +3}+5 k a_{k +2}-26 k a_{k +3}-\lambda a_{k +2}+2 \lambda a_{k +3}-\frac {5}{2} m a_{k +2}+13 m a_{k +3}+6 a_{k +2}-42 a_{k +3}}{4 k^{2}-4 k m +32 k -16 m +64} \\ \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 +4}=-\frac {a_{k} a^{2}-4 a^{2} a_{k +1}+5 a^{2} a_{k +2}-2 a^{2} a_{k +3}+k^{2} a_{k +2}-4 k^{2} a_{k +3}-k m a_{k +2}+4 k m a_{k +3}+\frac {1}{4} m^{2} a_{k +2}-m^{2} a_{k +3}+5 k a_{k +2}-26 k a_{k +3}-\lambda a_{k +2}+2 \lambda a_{k +3}-\frac {5}{2} m a_{k +2}+13 m a_{k +3}+6 a_{k +2}-42 a_{k +3}}{4 k^{2}-4 k m +32 k -16 m +64}, a_{1}=-\frac {2 a_{0} \left (a^{2}+\frac {1}{2} m^{2}-\lambda -\frac {1}{2} m \right )}{4 m -4}, a_{2}=\frac {a_{0} \left (4 a^{4}+4 a^{2} m^{2}+m^{4}-8 a^{2} \lambda +8 a^{2} m -4 \lambda \,m^{2}-5 m^{3}-8 a^{2}+4 \lambda ^{2}+8 \lambda m +8 m^{2}-8 \lambda -4 m \right )}{32 m^{2}-96 m +64}, a_{3}=-\frac {4 a_{0} \left (8 a^{2}-24 \lambda +31 m^{2}-12 m -24 a^{4}+2 a^{6}+\frac {55}{4} m^{4}+16 \lambda ^{2}-30 \lambda \,m^{2}+\frac {3}{2} a^{2} m^{3}+\frac {21}{2} \lambda \,m^{3}+15 a^{4} m -6 a^{2} \lambda m -3 m^{5}-6 a^{4} \lambda +3 a^{4} m^{2}+6 a^{2} \lambda ^{2}+\frac {3}{2} a^{2} m^{4}-30 m^{3}+\frac {1}{4} m^{6}-2 \lambda ^{3}-6 a^{2} \lambda \,m^{2}-\frac {3}{2} \lambda \,m^{4}-2 a^{2} m^{2}+3 \lambda ^{2} m^{2}+8 a^{2} \lambda -9 \lambda ^{2} m +36 \lambda m \right )}{384 m^{3}-2304 m^{2}+4224 m -2304}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +1 \\ {} & {} & \left [y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +1\right )^{k -\frac {m}{2}}, a_{k +4}=-\frac {a_{k} a^{2}-4 a^{2} a_{k +1}+5 a^{2} a_{k +2}-2 a^{2} a_{k +3}+k^{2} a_{k +2}-4 k^{2} a_{k +3}-k m a_{k +2}+4 k m a_{k +3}+\frac {1}{4} m^{2} a_{k +2}-m^{2} a_{k +3}+5 k a_{k +2}-26 k a_{k +3}-\lambda a_{k +2}+2 \lambda a_{k +3}-\frac {5}{2} m a_{k +2}+13 m a_{k +3}+6 a_{k +2}-42 a_{k +3}}{4 k^{2}-4 k m +32 k -16 m +64}, a_{1}=-\frac {2 a_{0} \left (a^{2}+\frac {1}{2} m^{2}-\lambda -\frac {1}{2} m \right )}{4 m -4}, a_{2}=\frac {a_{0} \left (4 a^{4}+4 a^{2} m^{2}+m^{4}-8 a^{2} \lambda +8 a^{2} m -4 \lambda \,m^{2}-5 m^{3}-8 a^{2}+4 \lambda ^{2}+8 \lambda m +8 m^{2}-8 \lambda -4 m \right )}{32 m^{2}-96 m +64}, a_{3}=-\frac {4 a_{0} \left (8 a^{2}-24 \lambda +31 m^{2}-12 m -24 a^{4}+2 a^{6}+\frac {55}{4} m^{4}+16 \lambda ^{2}-30 \lambda \,m^{2}+\frac {3}{2} a^{2} m^{3}+\frac {21}{2} \lambda \,m^{3}+15 a^{4} m -6 a^{2} \lambda m -3 m^{5}-6 a^{4} \lambda +3 a^{4} m^{2}+6 a^{2} \lambda ^{2}+\frac {3}{2} a^{2} m^{4}-30 m^{3}+\frac {1}{4} m^{6}-2 \lambda ^{3}-6 a^{2} \lambda \,m^{2}-\frac {3}{2} \lambda \,m^{4}-2 a^{2} m^{2}+3 \lambda ^{2} m^{2}+8 a^{2} \lambda -9 \lambda ^{2} m +36 \lambda m \right )}{384 m^{3}-2304 m^{2}+4224 m -2304}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {m}{2} \\ {} & {} & a_{k +4}=-\frac {a_{k} a^{2}-4 a^{2} a_{k +1}+5 a^{2} a_{k +2}-2 a^{2} a_{k +3}+k^{2} a_{k +2}-4 k^{2} a_{k +3}+k m a_{k +2}-4 k m a_{k +3}+\frac {1}{4} m^{2} a_{k +2}-m^{2} a_{k +3}+5 k a_{k +2}-26 k a_{k +3}-\lambda a_{k +2}+2 \lambda a_{k +3}+\frac {5}{2} m a_{k +2}-13 m a_{k +3}+6 a_{k +2}-42 a_{k +3}}{4 k^{2}+4 k m +32 k +16 m +64} \\ \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 +4}=-\frac {a_{k} a^{2}-4 a^{2} a_{k +1}+5 a^{2} a_{k +2}-2 a^{2} a_{k +3}+k^{2} a_{k +2}-4 k^{2} a_{k +3}+k m a_{k +2}-4 k m a_{k +3}+\frac {1}{4} m^{2} a_{k +2}-m^{2} a_{k +3}+5 k a_{k +2}-26 k a_{k +3}-\lambda a_{k +2}+2 \lambda a_{k +3}+\frac {5}{2} m a_{k +2}-13 m a_{k +3}+6 a_{k +2}-42 a_{k +3}}{4 k^{2}+4 k m +32 k +16 m +64}, a_{1}=-\frac {2 a_{0} \left (a^{2}+\frac {1}{2} m^{2}-\lambda +\frac {1}{2} m \right )}{-4 m -4}, a_{2}=\frac {a_{0} \left (4 a^{4}+4 a^{2} m^{2}+m^{4}-8 a^{2} \lambda -8 a^{2} m -4 \lambda \,m^{2}+5 m^{3}-8 a^{2}+4 \lambda ^{2}-8 \lambda m +8 m^{2}-8 \lambda +4 m \right )}{32 m^{2}+96 m +64}, a_{3}=-\frac {4 a_{0} \left (8 a^{2}-24 \lambda +31 m^{2}+12 m -24 a^{4}+2 a^{6}+\frac {55}{4} m^{4}+16 \lambda ^{2}-30 \lambda \,m^{2}-\frac {3}{2} a^{2} m^{3}-\frac {21}{2} \lambda \,m^{3}-15 a^{4} m +6 a^{2} \lambda m +3 m^{5}-6 a^{4} \lambda +3 a^{4} m^{2}+6 a^{2} \lambda ^{2}+\frac {3}{2} a^{2} m^{4}+30 m^{3}+\frac {1}{4} m^{6}-2 \lambda ^{3}-6 a^{2} \lambda \,m^{2}-\frac {3}{2} \lambda \,m^{4}-2 a^{2} m^{2}+3 \lambda ^{2} m^{2}+8 a^{2} \lambda +9 \lambda ^{2} m -36 \lambda m \right )}{-384 m^{3}-2304 m^{2}-4224 m -2304}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +1 \\ {} & {} & \left [y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +1\right )^{k +\frac {m}{2}}, a_{k +4}=-\frac {a_{k} a^{2}-4 a^{2} a_{k +1}+5 a^{2} a_{k +2}-2 a^{2} a_{k +3}+k^{2} a_{k +2}-4 k^{2} a_{k +3}+k m a_{k +2}-4 k m a_{k +3}+\frac {1}{4} m^{2} a_{k +2}-m^{2} a_{k +3}+5 k a_{k +2}-26 k a_{k +3}-\lambda a_{k +2}+2 \lambda a_{k +3}+\frac {5}{2} m a_{k +2}-13 m a_{k +3}+6 a_{k +2}-42 a_{k +3}}{4 k^{2}+4 k m +32 k +16 m +64}, a_{1}=-\frac {2 a_{0} \left (a^{2}+\frac {1}{2} m^{2}-\lambda +\frac {1}{2} m \right )}{-4 m -4}, a_{2}=\frac {a_{0} \left (4 a^{4}+4 a^{2} m^{2}+m^{4}-8 a^{2} \lambda -8 a^{2} m -4 \lambda \,m^{2}+5 m^{3}-8 a^{2}+4 \lambda ^{2}-8 \lambda m +8 m^{2}-8 \lambda +4 m \right )}{32 m^{2}+96 m +64}, a_{3}=-\frac {4 a_{0} \left (8 a^{2}-24 \lambda +31 m^{2}+12 m -24 a^{4}+2 a^{6}+\frac {55}{4} m^{4}+16 \lambda ^{2}-30 \lambda \,m^{2}-\frac {3}{2} a^{2} m^{3}-\frac {21}{2} \lambda \,m^{3}-15 a^{4} m +6 a^{2} \lambda m +3 m^{5}-6 a^{4} \lambda +3 a^{4} m^{2}+6 a^{2} \lambda ^{2}+\frac {3}{2} a^{2} m^{4}+30 m^{3}+\frac {1}{4} m^{6}-2 \lambda ^{3}-6 a^{2} \lambda \,m^{2}-\frac {3}{2} \lambda \,m^{4}-2 a^{2} m^{2}+3 \lambda ^{2} m^{2}+8 a^{2} \lambda +9 \lambda ^{2} m -36 \lambda m \right )}{-384 m^{3}-2304 m^{2}-4224 m -2304}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y \left (x \right )=\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 +4}=-\frac {b_{k} a^{2}-4 a^{2} b_{k +1}+5 a^{2} b_{k +2}-2 a^{2} b_{k +3}+k^{2} b_{k +2}-4 k^{2} b_{k +3}-k m b_{k +2}+4 k m b_{k +3}+\frac {1}{4} m^{2} b_{k +2}-m^{2} b_{k +3}+5 k b_{k +2}-26 k b_{k +3}-\lambda b_{k +2}+2 \lambda b_{k +3}-\frac {5}{2} m b_{k +2}+13 m b_{k +3}+6 b_{k +2}-42 b_{k +3}}{4 k^{2}-4 k m +32 k -16 m +64}, b_{1}=-\frac {2 b_{0} \left (a^{2}+\frac {1}{2} m^{2}-\lambda -\frac {1}{2} m \right )}{4 m -4}, b_{2}=\frac {b_{0} \left (4 a^{4}+4 a^{2} m^{2}+m^{4}-8 a^{2} \lambda +8 a^{2} m -4 \lambda \,m^{2}-5 m^{3}-8 a^{2}+4 \lambda ^{2}+8 \lambda m +8 m^{2}-8 \lambda -4 m \right )}{32 m^{2}-96 m +64}, b_{3}=-\frac {4 b_{0} \left (8 a^{2}-24 \lambda +31 m^{2}-12 m -24 a^{4}+2 a^{6}+\frac {55}{4} m^{4}+16 \lambda ^{2}-30 \lambda \,m^{2}+\frac {3}{2} a^{2} m^{3}+\frac {21}{2} \lambda \,m^{3}+15 a^{4} m -6 a^{2} \lambda m -3 m^{5}-6 a^{4} \lambda +3 a^{4} m^{2}+6 a^{2} \lambda ^{2}+\frac {3}{2} a^{2} m^{4}-30 m^{3}+\frac {1}{4} m^{6}-2 \lambda ^{3}-6 a^{2} \lambda \,m^{2}-\frac {3}{2} \lambda \,m^{4}-2 a^{2} m^{2}+3 \lambda ^{2} m^{2}+8 a^{2} \lambda -9 \lambda ^{2} m +36 \lambda m \right )}{384 m^{3}-2304 m^{2}+4224 m -2304}, c_{k +4}=-\frac {c_{k} a^{2}-4 a^{2} c_{k +1}+5 a^{2} c_{k +2}-2 a^{2} c_{k +3}+k^{2} c_{k +2}-4 k^{2} c_{k +3}+k m c_{k +2}-4 k m c_{k +3}+\frac {1}{4} m^{2} c_{k +2}-m^{2} c_{k +3}+5 k c_{k +2}-26 k c_{k +3}-\lambda c_{k +2}+2 \lambda c_{k +3}+\frac {5}{2} m c_{k +2}-13 m c_{k +3}+6 c_{k +2}-42 c_{k +3}}{4 k^{2}+4 k m +32 k +16 m +64}, c_{1}=-\frac {2 c_{0} \left (a^{2}+\frac {1}{2} m^{2}-\lambda +\frac {1}{2} m \right )}{-4 m -4}, c_{2}=\frac {c_{0} \left (4 a^{4}+4 a^{2} m^{2}+m^{4}-8 a^{2} \lambda -8 a^{2} m -4 \lambda \,m^{2}+5 m^{3}-8 a^{2}+4 \lambda ^{2}-8 \lambda m +8 m^{2}-8 \lambda +4 m \right )}{32 m^{2}+96 m +64}, c_{3}=-\frac {4 c_{0} \left (8 a^{2}-24 \lambda +31 m^{2}+12 m -24 a^{4}+2 a^{6}+\frac {55}{4} m^{4}+16 \lambda ^{2}-30 \lambda \,m^{2}-\frac {3}{2} a^{2} m^{3}-\frac {21}{2} \lambda \,m^{3}-15 a^{4} m +6 a^{2} \lambda m +3 m^{5}-6 a^{4} \lambda +3 a^{4} m^{2}+6 a^{2} \lambda ^{2}+\frac {3}{2} a^{2} m^{4}+30 m^{3}+\frac {1}{4} m^{6}-2 \lambda ^{3}-6 a^{2} \lambda \,m^{2}-\frac {3}{2} \lambda \,m^{4}-2 a^{2} m^{2}+3 \lambda ^{2} m^{2}+8 a^{2} \lambda +9 \lambda ^{2} m -36 \lambda m \right )}{-384 m^{3}-2304 m^{2}-4224 m -2304}\right ] \end {array} \]

2.33.26.2 ✓ Mathematica. Time used: 0.218 (sec). Leaf size: 234

ode=(x^2-1)^2*D[y[x],{x,2}]+2*x*(x^2-1)*D[y[x],x]+( (x^2-1)*(a^2*x^2-\[Lambda])-m^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} 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 ) \end{align*}

2.33.26.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
m = symbols("m") 
y = Function("y") 
ode = Eq(2*x*(x**2 - 1)*Derivative(y(x), x) + (-m**2 + (x**2 - 1)*(a**2*x**2 - lambda_))*y(x) + (x**2 - 1)**2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False