Internal problem ID [9689]
Internal file name [OUTPUT/8631_Monday_June_06_2022_04_32_10_AM_59677143/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1362.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_bessel_ode"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime } x^{2} \left (x^{2}-1\right )-2 x^{3} y^{\prime }-\left (\left (a -n \right ) \left (a +n +1\right ) x^{2} \left (x^{2}-1\right )+2 x^{2} a +n \left (1+n \right ) \left (x^{2}-1\right )\right ) y=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }-2 y^{\prime } x +\left (-x^{2} a^{2}+n^{2} x^{2}-x^{2} a +n \,x^{2}+a^{2}-2 n^{2}-a -2 n +\frac {n^{2}}{x^{2}}+\frac {n}{x^{2}}\right ) y = 0\tag {1} \end {align*}
Bessel ode has the form \begin {align*} x^{2} y^{\prime \prime }+y^{\prime } x +\left (-n^{2}+x^{2}\right ) y = 0\tag {2} \end {align*}
The generalized form of Bessel ode is given by Bowman (1958) as the following \begin {align*} x^{2} y^{\prime \prime }+\left (1-2 \alpha \right ) x y^{\prime }+\left (\beta ^{2} \gamma ^{2} x^{2 \gamma }-n^{2} \gamma ^{2}+\alpha ^{2}\right ) y = 0\tag {3} \end {align*}
With the standard solution \begin {align*} y&=x^{\alpha } \left (c_{1} \operatorname {BesselJ}\left (n , \beta \,x^{\gamma }\right )+c_{2} \operatorname {BesselY}\left (n , \beta \,x^{\gamma }\right )\right )\tag {4} \end {align*}
Comparing (3) to (1) and solving for \(\alpha ,\beta ,n,\gamma \) gives \begin {align*} \alpha &= {\frac {3}{2}}\\ \beta &= 2\\ n &= \sqrt {-4 a^{2}+8 n^{2}+4 a +8 n +9}\\ \gamma &= {\frac {1}{2}} \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = c_{1} x^{\frac {3}{2}} \operatorname {BesselJ}\left (\sqrt {-4 a^{2}+8 n^{2}+4 a +8 n +9}, 2 \sqrt {x}\right )+c_{2} x^{\frac {3}{2}} \operatorname {BesselY}\left (\sqrt {-4 a^{2}+8 n^{2}+4 a +8 n +9}, 2 \sqrt {x}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{\frac {3}{2}} \operatorname {BesselJ}\left (\sqrt {-4 a^{2}+8 n^{2}+4 a +8 n +9}, 2 \sqrt {x}\right )+c_{2} x^{\frac {3}{2}} \operatorname {BesselY}\left (\sqrt {-4 a^{2}+8 n^{2}+4 a +8 n +9}, 2 \sqrt {x}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{\frac {3}{2}} \operatorname {BesselJ}\left (\sqrt {-4 a^{2}+8 n^{2}+4 a +8 n +9}, 2 \sqrt {x}\right )+c_{2} x^{\frac {3}{2}} \operatorname {BesselY}\left (\sqrt {-4 a^{2}+8 n^{2}+4 a +8 n +9}, 2 \sqrt {x}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{4}-x^{2}\right ) y^{\prime \prime }-2 x^{3} y^{\prime }+\left (\left (-a^{2}+n^{2}-a +n \right ) x^{4}+\left (a^{2}-2 n^{2}-a -2 n \right ) x^{2}+n^{2}+n \right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=\frac {\left (a^{2} x^{4}-n^{2} x^{4}+x^{4} a -n \,x^{4}-x^{2} a^{2}+2 n^{2} x^{2}+x^{2} a +2 n \,x^{2}-n^{2}-n \right ) y}{x^{2} \left (x^{2}-1\right )}+\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} \\ {} & {} & y^{\prime \prime }-\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (a^{2} x^{4}-n^{2} x^{4}+x^{4} a -n \,x^{4}-x^{2} a^{2}+2 n^{2} x^{2}+x^{2} a +2 n \,x^{2}-n^{2}-n \right ) y}{x^{2} \left (x^{2}-1\right )}=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}-n^{2} x^{4}+x^{4} a -n \,x^{4}-x^{2} a^{2}+2 n^{2} x^{2}+x^{2} a +2 n \,x^{2}-n^{2}-n}{x^{2} \left (x^{2}-1\right )}\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}}}=0 \\ {} & \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} \\ {} & {} & y^{\prime \prime } x^{2} \left (x^{2}-1\right )-2 x^{3} y^{\prime }+\left (-a^{2} x^{4}+n^{2} x^{4}-x^{4} a +n \,x^{4}+x^{2} a^{2}-2 n^{2} x^{2}-x^{2} a -2 n \,x^{2}+n^{2}+n \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^{4}-4 u^{3}+5 u^{2}-2 u \right ) \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (-2 u^{3}+6 u^{2}-6 u +2\right ) \left (\frac {d}{d u}y \left (u \right )\right )+\left (-a^{2} u^{4}+n^{2} u^{4}+4 a^{2} u^{3}-a \,u^{4}-4 n^{2} u^{3}+n \,u^{4}-5 a^{2} u^{2}+4 a \,u^{3}+4 n^{2} u^{2}-4 n \,u^{3}+2 a^{2} u -7 a \,u^{2}+4 n \,u^{2}+6 a u -2 a \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 =0..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 =1..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} \\ {} & {} & -2 a_{0} r \left (-2+r \right ) u^{-1+r}+\left (-2 a_{1} \left (1+r \right ) \left (-1+r \right )-a_{0} \left (-5 r^{2}+2 a +11 r \right )\right ) u^{r}+\left (-2 a_{2} \left (2+r \right ) r -a_{1} \left (-5 r^{2}+2 a +r +6\right )+2 a_{0} \left (a^{2}-2 r^{2}+3 a +5 r \right )\right ) u^{1+r}+\left (-2 a_{3} \left (3+r \right ) \left (1+r \right )-a_{2} \left (-5 r^{2}+2 a -9 r +2\right )+2 a_{1} \left (a^{2}-2 r^{2}+3 a +r +3\right )-a_{0} \left (5 a^{2}-4 n^{2}-r^{2}+7 a -4 n +3 r \right )\right ) u^{2+r}+\left (-2 a_{4} \left (4+r \right ) \left (2+r \right )-a_{3} \left (-5 r^{2}+2 a -19 r -12\right )+2 a_{2} \left (a^{2}-2 r^{2}+3 a -3 r +2\right )-a_{1} \left (5 a^{2}-4 n^{2}-r^{2}+7 a -4 n +r +2\right )+4 a_{0} \left (a +n +1\right ) \left (a -n \right )\right ) u^{3+r}+\left (\moverset {\infty }{\munderset {k =4}{\sum }}\left (-2 a_{k +1} \left (k +1+r \right ) \left (k +r -1\right )-a_{k} \left (-5 k^{2}-10 k r -5 r^{2}+2 a +11 k +11 r \right )+2 a_{k -1} \left (a^{2}-2 \left (k -1\right )^{2}-4 \left (k -1\right ) r -2 r^{2}+3 a +5 k -5+5 r \right )-a_{k -2} \left (5 a^{2}-\left (k -2\right )^{2}-2 \left (k -2\right ) r -4 n^{2}-r^{2}+7 a +3 k -6-4 n +3 r \right )+4 a_{k -3} \left (a +n +1\right ) \left (a -n \right )-a_{k -4} \left (a +n +1\right ) \left (a -n \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -2 r \left (-2+r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, 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 (1+r \right ) \left (-1+r \right )-a_{0} \left (-5 r^{2}+2 a +11 r \right )=0, -2 a_{2} \left (2+r \right ) r -a_{1} \left (-5 r^{2}+2 a +r +6\right )+2 a_{0} \left (a^{2}-2 r^{2}+3 a +5 r \right )=0, -2 a_{3} \left (3+r \right ) \left (1+r \right )-a_{2} \left (-5 r^{2}+2 a -9 r +2\right )+2 a_{1} \left (a^{2}-2 r^{2}+3 a +r +3\right )-a_{0} \left (5 a^{2}-4 n^{2}-r^{2}+7 a -4 n +3 r \right )=0, -2 a_{4} \left (4+r \right ) \left (2+r \right )-a_{3} \left (-5 r^{2}+2 a -19 r -12\right )+2 a_{2} \left (a^{2}-2 r^{2}+3 a -3 r +2\right )-a_{1} \left (5 a^{2}-4 n^{2}-r^{2}+7 a -4 n +r +2\right )+4 a_{0} \left (a +n +1\right ) \left (a -n \right )=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{1}=-\frac {a_{0} \left (-5 r^{2}+2 a +11 r \right )}{2 \left (r^{2}-1\right )}, a_{2}=\frac {a_{0} \left (4 a^{2} r +17 r^{3}-8 a r -40 r^{2}+24 a -11 r +46\right )}{4 \left (r^{3}+2 r^{2}-r -2\right )}, a_{3}=-\frac {a_{0} \left (-20 a^{2} r^{2}-16 n^{2} r^{2}-49 r^{4}+16 a^{3}+28 a^{2} r +26 a \,r^{2}-16 n^{2} r -16 n \,r^{2}+72 r^{3}+56 a^{2}-110 a r +32 n^{2}-16 n r +173 r^{2}+132 a +32 n -288 r +92\right )}{8 \left (r^{4}+5 r^{3}+5 r^{2}-5 r -6\right )}, a_{4}=\frac {a_{0} \left (16 a^{4} r +68 a^{2} r^{3}+128 n^{2} r^{3}+129 r^{5}+16 a^{4}-64 a^{3} r -116 a^{2} r^{2}-64 a \,n^{2} r -76 a \,r^{3}+192 n^{2} r^{2}+128 n \,r^{3}+57 r^{4}+224 a^{3}-304 a^{2} r -64 a \,n^{2}-64 a n r +316 a \,r^{2}-608 n^{2} r +192 n \,r^{2}-903 r^{3}+576 a^{2}-64 a n -512 a r -96 n^{2}-608 n r +375 r^{2}+368 a -96 n +342 r \right )}{16 \left (r^{5}+9 r^{4}+25 r^{3}+15 r^{2}-26 r -24\right )}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (-2 a_{k +1}+5 a_{k}+a_{k -2}-4 a_{k -1}\right ) k^{2}+\left (2 \left (-2 a_{k +1}+5 a_{k}+a_{k -2}-4 a_{k -1}\right ) r -11 a_{k}-7 a_{k -2}+18 a_{k -1}\right ) k +\left (-2 a_{k +1}+5 a_{k}+a_{k -2}-4 a_{k -1}\right ) r^{2}+\left (-11 a_{k}-7 a_{k -2}+18 a_{k -1}\right ) r +\left (-5 a^{2}+4 n^{2}-7 a +4 n +10\right ) a_{k -2}+\left (-a_{k -4}+4 a_{k -3}+2 a_{k -1}\right ) a^{2}+\left (-2 a_{k}-a_{k -4}+4 a_{k -3}+6 a_{k -1}\right ) a -14 a_{k -1}+\left (a_{k -4}-4 a_{k -3}\right ) n^{2}+\left (a_{k -4}-4 a_{k -3}\right ) n +2 a_{k +1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +4 \\ {} & {} & \left (-2 a_{k +5}+5 a_{k +4}+a_{k +2}-4 a_{k +3}\right ) \left (k +4\right )^{2}+\left (2 \left (-2 a_{k +5}+5 a_{k +4}+a_{k +2}-4 a_{k +3}\right ) r -11 a_{k +4}-7 a_{k +2}+18 a_{k +3}\right ) \left (k +4\right )+\left (-2 a_{k +5}+5 a_{k +4}+a_{k +2}-4 a_{k +3}\right ) r^{2}+\left (-11 a_{k +4}-7 a_{k +2}+18 a_{k +3}\right ) r +\left (-5 a^{2}+4 n^{2}-7 a +4 n +10\right ) a_{k +2}+\left (-a_{k}+4 a_{k +1}+2 a_{k +3}\right ) a^{2}+\left (-2 a_{k +4}-a_{k}+4 a_{k +1}+6 a_{k +3}\right ) a -14 a_{k +3}+\left (a_{k}-4 a_{k +1}\right ) n^{2}+\left (a_{k}-4 a_{k +1}\right ) n +2 a_{k +5}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +5}=-\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}-5 k^{2} a_{k +4}-2 k r a_{k +2}+8 k r a_{k +3}-10 k r a_{k +4}-a_{k} n^{2}+4 n^{2} a_{k +1}-4 n^{2} a_{k +2}-r^{2} a_{k +2}+4 r^{2} a_{k +3}-5 r^{2} a_{k +4}+a_{k} a -4 a a_{k +1}+7 a a_{k +2}-6 a a_{k +3}+2 a a_{k +4}-k a_{k +2}+14 k a_{k +3}-29 k a_{k +4}-a_{k} n +4 n a_{k +1}-4 n a_{k +2}-r a_{k +2}+14 r a_{k +3}-29 r a_{k +4}+2 a_{k +2}+6 a_{k +3}-36 a_{k +4}}{2 \left (k^{2}+2 k r +r^{2}+8 k +8 r +15\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +5}=-\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}-5 k^{2} a_{k +4}-a_{k} n^{2}+4 n^{2} a_{k +1}-4 n^{2} a_{k +2}+a_{k} a -4 a a_{k +1}+7 a a_{k +2}-6 a a_{k +3}+2 a a_{k +4}-k a_{k +2}+14 k a_{k +3}-29 k a_{k +4}-a_{k} n +4 n a_{k +1}-4 n a_{k +2}+2 a_{k +2}+6 a_{k +3}-36 a_{k +4}}{2 \left (k^{2}+8 k +15\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k}, a_{k +5}=-\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}-5 k^{2} a_{k +4}-a_{k} n^{2}+4 n^{2} a_{k +1}-4 n^{2} a_{k +2}+a_{k} a -4 a a_{k +1}+7 a a_{k +2}-6 a a_{k +3}+2 a a_{k +4}-k a_{k +2}+14 k a_{k +3}-29 k a_{k +4}-a_{k} n +4 n a_{k +1}-4 n a_{k +2}+2 a_{k +2}+6 a_{k +3}-36 a_{k +4}}{2 \left (k^{2}+8 k +15\right )}, a_{1}=a_{0} a , a_{2}=-\frac {a_{0} \left (24 a +46\right )}{8}, a_{3}=\frac {a_{0} \left (16 a^{3}+56 a^{2}+32 n^{2}+132 a +32 n +92\right )}{48}, a_{4}=-\frac {a_{0} \left (16 a^{4}+224 a^{3}-64 a \,n^{2}+576 a^{2}-64 a n -96 n^{2}+368 a -96 n \right )}{384}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +1 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +1\right )^{k}, a_{k +5}=-\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}-5 k^{2} a_{k +4}-a_{k} n^{2}+4 n^{2} a_{k +1}-4 n^{2} a_{k +2}+a_{k} a -4 a a_{k +1}+7 a a_{k +2}-6 a a_{k +3}+2 a a_{k +4}-k a_{k +2}+14 k a_{k +3}-29 k a_{k +4}-a_{k} n +4 n a_{k +1}-4 n a_{k +2}+2 a_{k +2}+6 a_{k +3}-36 a_{k +4}}{2 \left (k^{2}+8 k +15\right )}, a_{1}=a_{0} a , a_{2}=-\frac {a_{0} \left (24 a +46\right )}{8}, a_{3}=\frac {a_{0} \left (16 a^{3}+56 a^{2}+32 n^{2}+132 a +32 n +92\right )}{48}, a_{4}=-\frac {a_{0} \left (16 a^{4}+224 a^{3}-64 a \,n^{2}+576 a^{2}-64 a n -96 n^{2}+368 a -96 n \right )}{384}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =2 \\ {} & {} & a_{k +5}=-\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}-5 k^{2} a_{k +4}-a_{k} n^{2}+4 n^{2} a_{k +1}-4 n^{2} a_{k +2}+a_{k} a -4 a a_{k +1}+7 a a_{k +2}-6 a a_{k +3}+2 a a_{k +4}-5 k a_{k +2}+30 k a_{k +3}-49 k a_{k +4}-a_{k} n +4 n a_{k +1}-4 n a_{k +2}-4 a_{k +2}+50 a_{k +3}-114 a_{k +4}}{2 \left (k^{2}+12 k +35\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =2 \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +2}, a_{k +5}=-\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}-5 k^{2} a_{k +4}-a_{k} n^{2}+4 n^{2} a_{k +1}-4 n^{2} a_{k +2}+a_{k} a -4 a a_{k +1}+7 a a_{k +2}-6 a a_{k +3}+2 a a_{k +4}-5 k a_{k +2}+30 k a_{k +3}-49 k a_{k +4}-a_{k} n +4 n a_{k +1}-4 n a_{k +2}-4 a_{k +2}+50 a_{k +3}-114 a_{k +4}}{2 \left (k^{2}+12 k +35\right )}, a_{1}=-\frac {a_{0} \left (2+2 a \right )}{6}, a_{2}=\frac {a_{0} \left (8 a^{2}+8 a \right )}{48}, a_{3}=-\frac {a_{0} \left (16 a^{3}+32 a^{2}-64 n^{2}+16 a -64 n \right )}{480}, a_{4}=\frac {a_{0} \left (48 a^{4}+96 a^{3}-192 a \,n^{2}+48 a^{2}-192 a n +480 n^{2}+480 n \right )}{5760}\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 +2}, a_{k +5}=-\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}-5 k^{2} a_{k +4}-a_{k} n^{2}+4 n^{2} a_{k +1}-4 n^{2} a_{k +2}+a_{k} a -4 a a_{k +1}+7 a a_{k +2}-6 a a_{k +3}+2 a a_{k +4}-5 k a_{k +2}+30 k a_{k +3}-49 k a_{k +4}-a_{k} n +4 n a_{k +1}-4 n a_{k +2}-4 a_{k +2}+50 a_{k +3}-114 a_{k +4}}{2 \left (k^{2}+12 k +35\right )}, a_{1}=-\frac {a_{0} \left (2+2 a \right )}{6}, a_{2}=\frac {a_{0} \left (8 a^{2}+8 a \right )}{48}, a_{3}=-\frac {a_{0} \left (16 a^{3}+32 a^{2}-64 n^{2}+16 a -64 n \right )}{480}, a_{4}=\frac {a_{0} \left (48 a^{4}+96 a^{3}-192 a \,n^{2}+48 a^{2}-192 a n +480 n^{2}+480 n \right )}{5760}\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}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}c_{k} \left (x +1\right )^{k +2}\right ), b_{k +5}=-\frac {a^{2} b_{k}-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}-5 k^{2} b_{k +4}-n^{2} b_{k}+4 n^{2} b_{k +1}-4 n^{2} b_{k +2}+a b_{k}-4 a b_{k +1}+7 a b_{k +2}-6 a b_{k +3}+2 a b_{k +4}-k b_{k +2}+14 k b_{k +3}-29 k b_{k +4}-n b_{k}+4 n b_{k +1}-4 n b_{k +2}+2 b_{k +2}+6 b_{k +3}-36 b_{k +4}}{2 \left (k^{2}+8 k +15\right )}, b_{1}=b_{0} a , b_{2}=-\frac {b_{0} \left (24 a +46\right )}{8}, b_{3}=\frac {b_{0} \left (16 a^{3}+56 a^{2}+32 n^{2}+132 a +32 n +92\right )}{48}, b_{4}=-\frac {b_{0} \left (16 a^{4}+224 a^{3}-64 a \,n^{2}+576 a^{2}-64 a n -96 n^{2}+368 a -96 n \right )}{384}, c_{k +5}=-\frac {a^{2} c_{k}-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}-5 k^{2} c_{k +4}-n^{2} c_{k}+4 n^{2} c_{k +1}-4 n^{2} c_{k +2}+a c_{k}-4 a c_{k +1}+7 a c_{k +2}-6 a c_{k +3}+2 a c_{k +4}-5 k c_{k +2}+30 k c_{k +3}-49 k c_{k +4}-n c_{k}+4 n c_{k +1}-4 n c_{k +2}-4 c_{k +2}+50 c_{k +3}-114 c_{k +4}}{2 \left (k^{2}+12 k +35\right )}, c_{1}=-\frac {c_{0} \left (2+2 a \right )}{6}, c_{2}=\frac {c_{0} \left (8 a^{2}+8 a \right )}{48}, c_{3}=-\frac {c_{0} \left (16 a^{3}+32 a^{2}-64 n^{2}+16 a -64 n \right )}{480}, c_{4}=\frac {c_{0} \left (48 a^{4}+96 a^{3}-192 a \,n^{2}+48 a^{2}-192 a n +480 n^{2}+480 n \right )}{5760}\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: 0.75 (sec). Leaf size: 109
dsolve(x^2*(x^2-1)*diff(diff(y(x),x),x)-2*x^3*diff(y(x),x)-((a-n)*(a+n+1)*x^2*(x^2-1)+2*a*x^2+n*(n+1)*(x^2-1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-n} \operatorname {HeunC}\left (0, -n -\frac {1}{2}, -2, -\frac {1}{4} a^{2}+\frac {1}{4} n^{2}-\frac {1}{4} a +\frac {1}{4} n , -\frac {1}{4} n^{2}-\frac {1}{4} n +\frac {3}{4}+\frac {1}{4} a^{2}-\frac {1}{4} a , x^{2}\right )+c_{2} x^{1+n} \operatorname {HeunC}\left (0, n +\frac {1}{2}, -2, -\frac {1}{4} a^{2}+\frac {1}{4} n^{2}-\frac {1}{4} a +\frac {1}{4} n , -\frac {1}{4} n^{2}-\frac {1}{4} n +\frac {3}{4}+\frac {1}{4} a^{2}-\frac {1}{4} a , x^{2}\right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x] == -(((2*a*x^2 + n*(1 + n)*(-1 + x^2) + (a - n)*(1 + a + n)*x^2*(-1 + x^2))*y[x])/(x^2*(-1 + x^2))) + (2*x*y'[x])/(-1 + x^2),y[x],x,IncludeSingularSolutions -> True]
Not solved