Internal problem ID [9698]
Internal file name [OUTPUT/8640_Monday_June_06_2022_04_34_38_AM_28627988/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1371.
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.
\[ \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}-1}+\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}}=0} \]
\[ \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 )+\left (2 x^{3}-2 x \right ) y^{\prime }+\left (-\lambda \,x^{2}-a^{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 (\lambda \,x^{2}+a^{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 (\lambda \,x^{2}+a^{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 {\lambda \,x^{2}+a^{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 {a^{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 (\lambda \,x^{2}+a^{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 (-\lambda \,u^{4}-a^{2} u^{2}+4 \lambda \,u^{3}+2 a^{2} u -4 \lambda \,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..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} \\ {} & {} & 2 a_{0} \left (a +2 r \right ) \left (a -2 r \right ) u^{1+r}+\left (2 a_{1} \left (a +2+2 r \right ) \left (a -2-2 r \right )-a_{0} \left (a^{2}-12 r^{2}+4 \lambda -4 r \right )\right ) u^{2+r}+\left (2 a_{2} \left (a +4+2 r \right ) \left (a -4-2 r \right )-a_{1} \left (a^{2}-12 r^{2}+4 \lambda -28 r -16\right )-2 a_{0} \left (3 r^{2}-2 \lambda +2 r \right )\right ) u^{3+r}+\left (\moverset {\infty }{\munderset {k =4}{\sum }}\left (2 a_{k -1} \left (a +2 k -2+2 r \right ) \left (a -2 k +2-2 r \right )-a_{k -2} \left (a^{2}-12 \left (k -2\right )^{2}-24 \left (k -2\right ) r -12 r^{2}-4 k +8+4 \lambda -4 r \right )-2 a_{k -3} \left (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 (\left (k -4\right )^{2}+2 \left (k -4\right ) r +r^{2}+k -4-\lambda +r \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & 2 \left (a +2 r \right ) \left (a -2 r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-\frac {a}{2}, \frac {a}{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 (a +2+2 r \right ) \left (a -2-2 r \right )-a_{0} \left (a^{2}-12 r^{2}+4 \lambda -4 r \right )=0, 2 a_{2} \left (a +4+2 r \right ) \left (a -4-2 r \right )-a_{1} \left (a^{2}-12 r^{2}+4 \lambda -28 r -16\right )-2 a_{0} \left (3 r^{2}-2 \lambda +2 r \right )=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{1}=\frac {a_{0} \left (a^{2}-12 r^{2}+4 \lambda -4 r \right )}{2 \left (a^{2}-4 r^{2}-8 r -4\right )}, a_{2}=\frac {a_{0} \left (a^{4}-12 a^{2} r^{2}+96 r^{4}-24 a^{2} r -64 \lambda \,r^{2}+256 r^{3}-16 a^{2}+16 \lambda ^{2}-64 \lambda r +192 r^{2}-32 \lambda +32 r \right )}{4 \left (a^{4}-8 a^{2} r^{2}+16 r^{4}-24 a^{2} r +96 r^{3}-20 a^{2}+208 r^{2}+192 r +64\right )}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (a_{k -4}-6 a_{k -3}+12 a_{k -2}-8 a_{k -1}\right ) k^{2}+\left (2 \left (a_{k -4}-6 a_{k -3}+12 a_{k -2}-8 a_{k -1}\right ) r -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 (-7 a_{k -4}+32 a_{k -3}-44 a_{k -2}+16 a_{k -1}\right ) r +\left (-a^{2}-4 \lambda +40\right ) a_{k -2}+\left (-\lambda +12\right ) a_{k -4}+2 a^{2} a_{k -1}+4 \lambda a_{k -3}-42 a_{k -3}-8 a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +4 \\ {} & {} & \left (a_{k}-6 a_{1+k}+12 a_{k +2}-8 a_{k +3}\right ) \left (k +4\right )^{2}+\left (2 \left (a_{k}-6 a_{1+k}+12 a_{k +2}-8 a_{k +3}\right ) r -7 a_{k}+32 a_{1+k}-44 a_{k +2}+16 a_{k +3}\right ) \left (k +4\right )+\left (a_{k}-6 a_{1+k}+12 a_{k +2}-8 a_{k +3}\right ) r^{2}+\left (-7 a_{k}+32 a_{1+k}-44 a_{k +2}+16 a_{k +3}\right ) r +\left (-a^{2}-4 \lambda +40\right ) a_{k +2}+\left (-\lambda +12\right ) a_{k}+2 a^{2} a_{k +3}+4 \lambda a_{1+k}-42 a_{1+k}-8 a_{k +3}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +3}=\frac {a^{2} a_{k +2}-k^{2} a_{k}+6 k^{2} a_{1+k}-12 k^{2} a_{k +2}-2 k r a_{k}+12 k r a_{1+k}-24 k r a_{k +2}-r^{2} a_{k}+6 r^{2} a_{1+k}-12 r^{2} a_{k +2}-k a_{k}+16 k a_{1+k}-52 k a_{k +2}+a_{k} \lambda -4 \lambda a_{1+k}+4 \lambda a_{k +2}-r a_{k}+16 r a_{1+k}-52 r a_{k +2}+10 a_{1+k}-56 a_{k +2}}{2 \left (a^{2}-4 k^{2}-8 k r -4 r^{2}-24 k -24 r -36\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {a}{2} \\ {} & {} & a_{k +3}=\frac {-\frac {1}{4} a_{k} a^{2}+\frac {3}{2} a^{2} a_{1+k}-2 a^{2} a_{k +2}+a k a_{k}-6 a k a_{1+k}+12 a k a_{k +2}-k^{2} a_{k}+6 k^{2} a_{1+k}-12 k^{2} a_{k +2}+\frac {1}{2} a a_{k}-8 a a_{1+k}+26 a a_{k +2}-k a_{k}+16 k a_{1+k}-52 k a_{k +2}+a_{k} \lambda -4 \lambda a_{1+k}+4 \lambda a_{k +2}+10 a_{1+k}-56 a_{k +2}}{2 \left (4 a k -4 k^{2}+12 a -24 k -36\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {a}{2} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k -\frac {a}{2}}, a_{k +3}=\frac {-\frac {1}{4} a_{k} a^{2}+\frac {3}{2} a^{2} a_{1+k}-2 a^{2} a_{k +2}+a k a_{k}-6 a k a_{1+k}+12 a k a_{k +2}-k^{2} a_{k}+6 k^{2} a_{1+k}-12 k^{2} a_{k +2}+\frac {1}{2} a a_{k}-8 a a_{1+k}+26 a a_{k +2}-k a_{k}+16 k a_{1+k}-52 k a_{k +2}+a_{k} \lambda -4 \lambda a_{1+k}+4 \lambda a_{k +2}+10 a_{1+k}-56 a_{k +2}}{2 \left (4 a k -4 k^{2}+12 a -24 k -36\right )}, a_{1}=\frac {a_{0} \left (-2 a^{2}+2 a +4 \lambda \right )}{2 \left (4 a -4\right )}, a_{2}=\frac {a_{0} \left (4 a^{4}-20 a^{3}-16 a^{2} \lambda +32 a^{2}+32 a \lambda +16 \lambda ^{2}-16 a -32 \lambda \right )}{4 \left (32 a^{2}-96 a +64\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 {a}{2}}, a_{k +3}=\frac {-\frac {1}{4} a_{k} a^{2}+\frac {3}{2} a^{2} a_{1+k}-2 a^{2} a_{k +2}+a k a_{k}-6 a k a_{1+k}+12 a k a_{k +2}-k^{2} a_{k}+6 k^{2} a_{1+k}-12 k^{2} a_{k +2}+\frac {1}{2} a a_{k}-8 a a_{1+k}+26 a a_{k +2}-k a_{k}+16 k a_{1+k}-52 k a_{k +2}+a_{k} \lambda -4 \lambda a_{1+k}+4 \lambda a_{k +2}+10 a_{1+k}-56 a_{k +2}}{2 \left (4 a k -4 k^{2}+12 a -24 k -36\right )}, a_{1}=\frac {a_{0} \left (-2 a^{2}+2 a +4 \lambda \right )}{2 \left (4 a -4\right )}, a_{2}=\frac {a_{0} \left (4 a^{4}-20 a^{3}-16 a^{2} \lambda +32 a^{2}+32 a \lambda +16 \lambda ^{2}-16 a -32 \lambda \right )}{4 \left (32 a^{2}-96 a +64\right )}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {a}{2} \\ {} & {} & a_{k +3}=\frac {-\frac {1}{4} a_{k} a^{2}+\frac {3}{2} a^{2} a_{1+k}-2 a^{2} a_{k +2}-a k a_{k}+6 a k a_{1+k}-12 a k a_{k +2}-k^{2} a_{k}+6 k^{2} a_{1+k}-12 k^{2} a_{k +2}-\frac {1}{2} a a_{k}+8 a a_{1+k}-26 a a_{k +2}-k a_{k}+16 k a_{1+k}-52 k a_{k +2}+a_{k} \lambda -4 \lambda a_{1+k}+4 \lambda a_{k +2}+10 a_{1+k}-56 a_{k +2}}{2 \left (-4 a k -4 k^{2}-12 a -24 k -36\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {a}{2} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +\frac {a}{2}}, a_{k +3}=\frac {-\frac {1}{4} a_{k} a^{2}+\frac {3}{2} a^{2} a_{1+k}-2 a^{2} a_{k +2}-a k a_{k}+6 a k a_{1+k}-12 a k a_{k +2}-k^{2} a_{k}+6 k^{2} a_{1+k}-12 k^{2} a_{k +2}-\frac {1}{2} a a_{k}+8 a a_{1+k}-26 a a_{k +2}-k a_{k}+16 k a_{1+k}-52 k a_{k +2}+a_{k} \lambda -4 \lambda a_{1+k}+4 \lambda a_{k +2}+10 a_{1+k}-56 a_{k +2}}{2 \left (-4 a k -4 k^{2}-12 a -24 k -36\right )}, a_{1}=\frac {a_{0} \left (-2 a^{2}-2 a +4 \lambda \right )}{2 \left (-4 a -4\right )}, a_{2}=\frac {a_{0} \left (4 a^{4}+20 a^{3}-16 a^{2} \lambda +32 a^{2}-32 a \lambda +16 \lambda ^{2}+16 a -32 \lambda \right )}{4 \left (32 a^{2}+96 a +64\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 {a}{2}}, a_{k +3}=\frac {-\frac {1}{4} a_{k} a^{2}+\frac {3}{2} a^{2} a_{1+k}-2 a^{2} a_{k +2}-a k a_{k}+6 a k a_{1+k}-12 a k a_{k +2}-k^{2} a_{k}+6 k^{2} a_{1+k}-12 k^{2} a_{k +2}-\frac {1}{2} a a_{k}+8 a a_{1+k}-26 a a_{k +2}-k a_{k}+16 k a_{1+k}-52 k a_{k +2}+a_{k} \lambda -4 \lambda a_{1+k}+4 \lambda a_{k +2}+10 a_{1+k}-56 a_{k +2}}{2 \left (-4 a k -4 k^{2}-12 a -24 k -36\right )}, a_{1}=\frac {a_{0} \left (-2 a^{2}-2 a +4 \lambda \right )}{2 \left (-4 a -4\right )}, a_{2}=\frac {a_{0} \left (4 a^{4}+20 a^{3}-16 a^{2} \lambda +32 a^{2}-32 a \lambda +16 \lambda ^{2}+16 a -32 \lambda \right )}{4 \left (32 a^{2}+96 a +64\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 {a}{2}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}c_{k} \left (x +1\right )^{k +\frac {a}{2}}\right ), b_{k +3}=\frac {-\frac {1}{4} b_{k} a^{2}+\frac {3}{2} a^{2} b_{k +1}-2 a^{2} b_{k +2}+a k b_{k}-6 a k b_{k +1}+12 a k b_{k +2}-k^{2} b_{k}+6 k^{2} b_{k +1}-12 k^{2} b_{k +2}+\frac {1}{2} a b_{k}-8 a b_{k +1}+26 a b_{k +2}-k b_{k}+16 k b_{k +1}-52 k b_{k +2}+b_{k} \lambda -4 \lambda b_{k +1}+4 \lambda b_{k +2}+10 b_{k +1}-56 b_{k +2}}{2 \left (4 a k -4 k^{2}+12 a -24 k -36\right )}, b_{1}=\frac {b_{0} \left (-2 a^{2}+2 a +4 \lambda \right )}{2 \left (4 a -4\right )}, b_{2}=\frac {b_{0} \left (4 a^{4}-20 a^{3}-16 a^{2} \lambda +32 a^{2}+32 a \lambda +16 \lambda ^{2}-16 a -32 \lambda \right )}{4 \left (32 a^{2}-96 a +64\right )}, c_{k +3}=\frac {-\frac {1}{4} c_{k} a^{2}+\frac {3}{2} a^{2} c_{k +1}-2 a^{2} c_{k +2}-a k c_{k}+6 a k c_{k +1}-12 a k c_{k +2}-k^{2} c_{k}+6 k^{2} c_{k +1}-12 k^{2} c_{k +2}-\frac {1}{2} a c_{k}+8 a c_{k +1}-26 a c_{k +2}-k c_{k}+16 k c_{k +1}-52 k c_{k +2}+c_{k} \lambda -4 \lambda c_{k +1}+4 \lambda c_{k +2}+10 c_{k +1}-56 c_{k +2}}{2 \left (-4 a k -4 k^{2}-12 a -24 k -36\right )}, c_{1}=\frac {c_{0} \left (-2 a^{2}-2 a +4 \lambda \right )}{2 \left (-4 a -4\right )}, c_{2}=\frac {c_{0} \left (4 a^{4}+20 a^{3}-16 a^{2} \lambda +32 a^{2}-32 a \lambda +16 \lambda ^{2}+16 a -32 \lambda \right )}{4 \left (32 a^{2}+96 a +64\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: 0.094 (sec). Leaf size: 37
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)-(-a^2-lambda*(x^2-1))/(x^2-1)^2*y(x),y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {LegendreP}\left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right )+c_{2} \operatorname {LegendreQ}\left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right ) \]
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 48
DSolve[y''[x] == -(((-a^2 - \[Lambda]*(-1 + x^2))*y[x])/(-1 + x^2)^2) - (2*x*y'[x])/(-1 + x^2),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 P_{\frac {1}{2} \left (\sqrt {4 \lambda +1}-1\right )}^a(x)+c_2 Q_{\frac {1}{2} \left (\sqrt {4 \lambda +1}-1\right )}^a(x) \]