Internal problem ID [9894]
Internal file name [OUTPUT/8841_Monday_June_06_2022_05_36_41_AM_25362123/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1572.
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_high_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {\left (x^{2}-1\right )^{2} y^{\prime \prime \prime \prime }+10 x \left (x^{2}-1\right ) y^{\prime \prime \prime }+\left (24 x^{2}-8-2 \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )\right ) \left (x^{2}-1\right )\right ) y^{\prime \prime }-6 x \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )-2\right ) y^{\prime }+\left (\left (\mu \left (\mu +1\right )-\nu \left (\nu +1\right )\right )^{2}-2 \mu \left (\mu +1\right )-2 \nu \left (\nu +1\right )\right ) y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{2}-1\right )^{2} \left (\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\right )+10 x \left (x^{2}-1\right ) \left (\frac {d}{d x}y^{\prime \prime }\right )+\left (24 x^{2}-8-2 \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )\right ) \left (x^{2}-1\right )\right ) \left (\frac {d}{d x}y^{\prime }\right )-6 x \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )-2\right ) y^{\prime }+\left (\left (\mu \left (\mu +1\right )-\nu \left (\nu +1\right )\right )^{2}-2 \mu \left (\mu +1\right )-2 \nu \left (\nu +1\right )\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime } \\ \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 {10 x}{x^{2}-1}, P_{3}\left (x \right )=-\frac {2 \left (\mu ^{2} x^{2}+\nu ^{2} x^{2}+\mu \,x^{2}+\nu \,x^{2}-\mu ^{2}-\nu ^{2}-12 x^{2}-\mu -\nu +4\right )}{\left (x^{2}-1\right )^{2}}, P_{4}\left (x \right )=-\frac {6 x \left (\mu ^{2}+\nu ^{2}+\mu +\nu -2\right )}{\left (x^{2}-1\right )^{2}}, P_{5}\left (x \right )=\frac {\mu ^{4}-2 \nu ^{2} \mu ^{2}+\nu ^{4}+2 \mu ^{3}-2 \nu \,\mu ^{2}-2 \nu ^{2} \mu +2 \nu ^{3}-\mu ^{2}-2 \nu \mu -\nu ^{2}-2 \mu -2 \nu }{\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}}}=5 \\ {} & \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}}}=4 \\ {} & \circ & \left (x +1\right )^{3}\cdot P_{4}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-1 \\ {} & {} & \left (\left (x +1\right )^{3}\cdot P_{4}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-1}}}=0 \\ {} & \circ & \left (x +1\right )^{4}\cdot P_{5}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-1 \\ {} & {} & \left (\left (x +1\right )^{4}\cdot P_{5}\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} \\ {} & {} & \left (x^{2}-1\right )^{2} \left (\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\right )+10 x \left (x^{2}-1\right ) \left (\frac {d}{d x}y^{\prime \prime }\right )+\left (-2 \mu ^{2} x^{2}-2 \nu ^{2} x^{2}-2 \mu \,x^{2}-2 \nu \,x^{2}+2 \mu ^{2}+2 \nu ^{2}+24 x^{2}+2 \mu +2 \nu -8\right ) \left (\frac {d}{d x}y^{\prime }\right )-6 x \left (\mu ^{2}+\nu ^{2}+\mu +\nu -2\right ) y^{\prime }+\left (\mu ^{4}-2 \nu ^{2} \mu ^{2}+\nu ^{4}+2 \mu ^{3}-2 \nu \,\mu ^{2}-2 \nu ^{2} \mu +2 \nu ^{3}-\mu ^{2}-2 \nu \mu -\nu ^{2}-2 \mu -2 \nu \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}+4 u^{2}\right ) \left (\frac {d}{d u}\frac {d}{d u}\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (10 u^{3}-30 u^{2}+20 u \right ) \left (\frac {d}{d u}\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (-2 \mu ^{2} u^{2}-2 \nu ^{2} u^{2}+4 \mu ^{2} u -2 \mu \,u^{2}+4 \nu ^{2} u -2 \nu \,u^{2}+4 \mu u +4 \nu u +24 u^{2}-48 u +16\right ) \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )+\left (-6 \mu ^{2} u -6 \nu ^{2} u +6 \mu ^{2}-6 \mu u +6 \nu ^{2}-6 \nu u +6 \mu +6 \nu +12 u -12\right ) \left (\frac {d}{d u}y \left (u \right )\right )+\left (\mu ^{4}-2 \nu ^{2} \mu ^{2}+\nu ^{4}+2 \mu ^{3}-2 \nu \,\mu ^{2}-2 \nu ^{2} \mu +2 \nu ^{3}-\mu ^{2}-2 \nu \mu -\nu ^{2}-2 \mu -2 \nu \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 \left (\frac {d}{d u}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & 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 =0..2 \\ {} & {} & 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} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot \left (\frac {d}{d u}\frac {d^{2}}{d u^{2}}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}\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 ) \left (k +r -2\right ) u^{k +r -3+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +3-m \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}\frac {d^{2}}{d u^{2}}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-3+m}{\sum }}a_{k +3-m} \left (k +3-m +r \right ) \left (k +2-m +r \right ) \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}\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}{d u}\frac {d}{d u}\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 ) \left (k +r -2\right ) \left (k +r -3\right ) u^{k +r -4+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +4-m \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}\frac {d}{d u}\frac {d^{2}}{d u^{2}}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-4+m}{\sum }}a_{k +4-m} \left (k +4-m +r \right ) \left (k +3-m +r \right ) \left (k +2-m +r \right ) \left (k +1-m +r \right ) u^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & 4 a_{0} r^{3} \left (-1+r \right ) u^{-2+r}+\left (4 a_{1} \left (1+r \right )^{3} r -2 a_{0} \left (1+2 r \right ) r \left (-\mu ^{2}-\nu ^{2}+r^{2}-\mu -\nu +r \right )\right ) u^{-1+r}+\left (\moverset {\infty }{\munderset {k =0}{\sum }}\left (4 a_{k +2} \left (k +2+r \right )^{3} \left (k +1+r \right )-2 a_{k +1} \left (2 k +3+2 r \right ) \left (k +1+r \right ) \left (\left (k +1\right )^{2}+2 \left (k +1\right ) r -\mu ^{2}-\nu ^{2}+r^{2}+k +1-\mu -\nu +r \right )+a_{k} \left (k +\mu +\nu +r +2\right ) \left (r +1+\nu -\mu +k \right ) \left (r +1-\nu +\mu +k \right ) \left (r -\nu -\mu +k \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & 4 r^{3} \left (-1+r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, 1\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & 4 a_{k +2} \left (k +2+r \right )^{3} \left (k +1+r \right )+a_{k} \left (k +\mu +\nu +r +2\right ) \left (r +1+\nu -\mu +k \right ) \left (r +1-\nu +\mu +k \right ) \left (r -\nu -\mu +k \right )-4 a_{k +1} \left (k^{2}+\left (2 r +3\right ) k +r^{2}-\mu ^{2}-\nu ^{2}+3 r -\mu -\nu +2\right ) \left (k +\frac {3}{2}+r \right ) \left (k +1+r \right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {k^{4} a_{k}-4 k^{4} a_{k +1}+4 k^{3} r a_{k}-16 k^{3} r a_{k +1}-2 k^{2} \mu ^{2} a_{k}+4 k^{2} \mu ^{2} a_{k +1}-2 k^{2} \nu ^{2} a_{k}+4 k^{2} \nu ^{2} a_{k +1}+6 k^{2} r^{2} a_{k}-24 k^{2} r^{2} a_{k +1}-4 k \,\mu ^{2} r a_{k}+8 k \,\mu ^{2} r a_{k +1}-4 k \,\nu ^{2} r a_{k}+8 k \,\nu ^{2} r a_{k +1}+4 k \,r^{3} a_{k}-16 k \,r^{3} a_{k +1}+a_{k} \mu ^{4}-2 a_{k} \nu ^{2} \mu ^{2}-2 \mu ^{2} r^{2} a_{k}+4 \mu ^{2} r^{2} a_{k +1}+a_{k} \nu ^{4}-2 \nu ^{2} r^{2} a_{k}+4 \nu ^{2} r^{2} a_{k +1}+r^{4} a_{k}-4 r^{4} a_{k +1}+4 k^{3} a_{k}-22 k^{3} a_{k +1}-2 k^{2} \mu a_{k}+4 k^{2} \mu a_{k +1}-2 k^{2} \nu a_{k}+4 k^{2} \nu a_{k +1}+12 k^{2} r a_{k}-66 k^{2} r a_{k +1}-4 k \,\mu ^{2} a_{k}+10 k \,\mu ^{2} a_{k +1}-4 k \mu r a_{k}+8 k \mu r a_{k +1}-4 k \,\nu ^{2} a_{k}+10 k \,\nu ^{2} a_{k +1}-4 k \nu r a_{k}+8 k \nu r a_{k +1}+12 k \,r^{2} a_{k}-66 k \,r^{2} a_{k +1}+2 a_{k} \mu ^{3}-2 a_{k} \nu \,\mu ^{2}-4 \mu ^{2} r a_{k}+10 \mu ^{2} r a_{k +1}-2 a_{k} \nu ^{2} \mu -2 \mu \,r^{2} a_{k}+4 \mu \,r^{2} a_{k +1}+2 a_{k} \nu ^{3}-4 \nu ^{2} r a_{k}+10 \nu ^{2} r a_{k +1}-2 \nu \,r^{2} a_{k}+4 \nu \,r^{2} a_{k +1}+4 r^{3} a_{k}-22 r^{3} a_{k +1}+5 k^{2} a_{k}-44 k^{2} a_{k +1}-4 k \mu a_{k}+10 k \mu a_{k +1}-4 k \nu a_{k}+10 k \nu a_{k +1}+10 k r a_{k}-88 k r a_{k +1}-a_{k} \mu ^{2}+6 \mu ^{2} a_{k +1}-2 a_{k} \nu \mu -4 \mu r a_{k}+10 \mu r a_{k +1}-a_{k} \nu ^{2}+6 \nu ^{2} a_{k +1}-4 \nu r a_{k}+10 \nu r a_{k +1}+5 r^{2} a_{k}-44 r^{2} a_{k +1}+2 k a_{k}-38 k a_{k +1}-2 a_{k} \mu +6 \mu a_{k +1}-2 a_{k} \nu +6 \nu a_{k +1}+2 r a_{k}-38 r a_{k +1}-12 a_{k +1}}{4 \left (k +2+r \right )^{3} \left (k +1+r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +2}=-\frac {k^{4} a_{k}-4 k^{4} a_{k +1}-2 k^{2} \mu ^{2} a_{k}+4 k^{2} \mu ^{2} a_{k +1}-2 k^{2} \nu ^{2} a_{k}+4 k^{2} \nu ^{2} a_{k +1}+a_{k} \mu ^{4}-2 a_{k} \nu ^{2} \mu ^{2}+a_{k} \nu ^{4}+4 k^{3} a_{k}-22 k^{3} a_{k +1}-2 k^{2} \mu a_{k}+4 k^{2} \mu a_{k +1}-2 k^{2} \nu a_{k}+4 k^{2} \nu a_{k +1}-4 k \,\mu ^{2} a_{k}+10 k \,\mu ^{2} a_{k +1}-4 k \,\nu ^{2} a_{k}+10 k \,\nu ^{2} a_{k +1}+2 a_{k} \mu ^{3}-2 a_{k} \nu \,\mu ^{2}-2 a_{k} \nu ^{2} \mu +2 a_{k} \nu ^{3}+5 k^{2} a_{k}-44 k^{2} a_{k +1}-4 k \mu a_{k}+10 k \mu a_{k +1}-4 k \nu a_{k}+10 k \nu a_{k +1}-a_{k} \mu ^{2}+6 \mu ^{2} a_{k +1}-2 a_{k} \nu \mu -a_{k} \nu ^{2}+6 \nu ^{2} a_{k +1}+2 k a_{k}-38 k a_{k +1}-2 a_{k} \mu +6 \mu a_{k +1}-2 a_{k} \nu +6 \nu a_{k +1}-12 a_{k +1}}{4 \left (k +2\right )^{3} \left (k +1\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 +2}=-\frac {k^{4} a_{k}-4 k^{4} a_{k +1}-2 k^{2} \mu ^{2} a_{k}+4 k^{2} \mu ^{2} a_{k +1}-2 k^{2} \nu ^{2} a_{k}+4 k^{2} \nu ^{2} a_{k +1}+a_{k} \mu ^{4}-2 a_{k} \nu ^{2} \mu ^{2}+a_{k} \nu ^{4}+4 k^{3} a_{k}-22 k^{3} a_{k +1}-2 k^{2} \mu a_{k}+4 k^{2} \mu a_{k +1}-2 k^{2} \nu a_{k}+4 k^{2} \nu a_{k +1}-4 k \,\mu ^{2} a_{k}+10 k \,\mu ^{2} a_{k +1}-4 k \,\nu ^{2} a_{k}+10 k \,\nu ^{2} a_{k +1}+2 a_{k} \mu ^{3}-2 a_{k} \nu \,\mu ^{2}-2 a_{k} \nu ^{2} \mu +2 a_{k} \nu ^{3}+5 k^{2} a_{k}-44 k^{2} a_{k +1}-4 k \mu a_{k}+10 k \mu a_{k +1}-4 k \nu a_{k}+10 k \nu a_{k +1}-a_{k} \mu ^{2}+6 \mu ^{2} a_{k +1}-2 a_{k} \nu \mu -a_{k} \nu ^{2}+6 \nu ^{2} a_{k +1}+2 k a_{k}-38 k a_{k +1}-2 a_{k} \mu +6 \mu a_{k +1}-2 a_{k} \nu +6 \nu a_{k +1}-12 a_{k +1}}{4 \left (k +2\right )^{3} \left (k +1\right )}, 0=0\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 +2}=-\frac {k^{4} a_{k}-4 k^{4} a_{k +1}-2 k^{2} \mu ^{2} a_{k}+4 k^{2} \mu ^{2} a_{k +1}-2 k^{2} \nu ^{2} a_{k}+4 k^{2} \nu ^{2} a_{k +1}+a_{k} \mu ^{4}-2 a_{k} \nu ^{2} \mu ^{2}+a_{k} \nu ^{4}+4 k^{3} a_{k}-22 k^{3} a_{k +1}-2 k^{2} \mu a_{k}+4 k^{2} \mu a_{k +1}-2 k^{2} \nu a_{k}+4 k^{2} \nu a_{k +1}-4 k \,\mu ^{2} a_{k}+10 k \,\mu ^{2} a_{k +1}-4 k \,\nu ^{2} a_{k}+10 k \,\nu ^{2} a_{k +1}+2 a_{k} \mu ^{3}-2 a_{k} \nu \,\mu ^{2}-2 a_{k} \nu ^{2} \mu +2 a_{k} \nu ^{3}+5 k^{2} a_{k}-44 k^{2} a_{k +1}-4 k \mu a_{k}+10 k \mu a_{k +1}-4 k \nu a_{k}+10 k \nu a_{k +1}-a_{k} \mu ^{2}+6 \mu ^{2} a_{k +1}-2 a_{k} \nu \mu -a_{k} \nu ^{2}+6 \nu ^{2} a_{k +1}+2 k a_{k}-38 k a_{k +1}-2 a_{k} \mu +6 \mu a_{k +1}-2 a_{k} \nu +6 \nu a_{k +1}-12 a_{k +1}}{4 \left (k +2\right )^{3} \left (k +1\right )}, 0=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =1 \\ {} & {} & a_{k +2}=-\frac {k^{4} a_{k}-4 k^{4} a_{k +1}-2 k^{2} \mu ^{2} a_{k}+4 k^{2} \mu ^{2} a_{k +1}-2 k^{2} \nu ^{2} a_{k}+4 k^{2} \nu ^{2} a_{k +1}+a_{k} \mu ^{4}-2 a_{k} \nu ^{2} \mu ^{2}+a_{k} \nu ^{4}+8 k^{3} a_{k}-38 k^{3} a_{k +1}-2 k^{2} \mu a_{k}+4 k^{2} \mu a_{k +1}-2 k^{2} \nu a_{k}+4 k^{2} \nu a_{k +1}-8 k \,\mu ^{2} a_{k}+18 k \,\mu ^{2} a_{k +1}-8 k \,\nu ^{2} a_{k}+18 k \,\nu ^{2} a_{k +1}+2 a_{k} \mu ^{3}-2 a_{k} \nu \,\mu ^{2}-2 a_{k} \nu ^{2} \mu +2 a_{k} \nu ^{3}+23 k^{2} a_{k}-134 k^{2} a_{k +1}-8 k \mu a_{k}+18 k \mu a_{k +1}-8 k \nu a_{k}+18 k \nu a_{k +1}-7 a_{k} \mu ^{2}+20 \mu ^{2} a_{k +1}-2 a_{k} \nu \mu -7 a_{k} \nu ^{2}+20 \nu ^{2} a_{k +1}+28 k a_{k}-208 k a_{k +1}-8 a_{k} \mu +20 \mu a_{k +1}-8 a_{k} \nu +20 \nu a_{k +1}+12 a_{k}-120 a_{k +1}}{4 \left (k +3\right )^{3} \left (k +2\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =1 \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +1}, a_{k +2}=-\frac {k^{4} a_{k}-4 k^{4} a_{k +1}-2 k^{2} \mu ^{2} a_{k}+4 k^{2} \mu ^{2} a_{k +1}-2 k^{2} \nu ^{2} a_{k}+4 k^{2} \nu ^{2} a_{k +1}+a_{k} \mu ^{4}-2 a_{k} \nu ^{2} \mu ^{2}+a_{k} \nu ^{4}+8 k^{3} a_{k}-38 k^{3} a_{k +1}-2 k^{2} \mu a_{k}+4 k^{2} \mu a_{k +1}-2 k^{2} \nu a_{k}+4 k^{2} \nu a_{k +1}-8 k \,\mu ^{2} a_{k}+18 k \,\mu ^{2} a_{k +1}-8 k \,\nu ^{2} a_{k}+18 k \,\nu ^{2} a_{k +1}+2 a_{k} \mu ^{3}-2 a_{k} \nu \,\mu ^{2}-2 a_{k} \nu ^{2} \mu +2 a_{k} \nu ^{3}+23 k^{2} a_{k}-134 k^{2} a_{k +1}-8 k \mu a_{k}+18 k \mu a_{k +1}-8 k \nu a_{k}+18 k \nu a_{k +1}-7 a_{k} \mu ^{2}+20 \mu ^{2} a_{k +1}-2 a_{k} \nu \mu -7 a_{k} \nu ^{2}+20 \nu ^{2} a_{k +1}+28 k a_{k}-208 k a_{k +1}-8 a_{k} \mu +20 \mu a_{k +1}-8 a_{k} \nu +20 \nu a_{k +1}+12 a_{k}-120 a_{k +1}}{4 \left (k +3\right )^{3} \left (k +2\right )}, 32 a_{1}-6 a_{0} \left (-\mu ^{2}-\nu ^{2}-\mu -\nu +2\right )=0\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 +1}, a_{k +2}=-\frac {k^{4} a_{k}-4 k^{4} a_{k +1}-2 k^{2} \mu ^{2} a_{k}+4 k^{2} \mu ^{2} a_{k +1}-2 k^{2} \nu ^{2} a_{k}+4 k^{2} \nu ^{2} a_{k +1}+a_{k} \mu ^{4}-2 a_{k} \nu ^{2} \mu ^{2}+a_{k} \nu ^{4}+8 k^{3} a_{k}-38 k^{3} a_{k +1}-2 k^{2} \mu a_{k}+4 k^{2} \mu a_{k +1}-2 k^{2} \nu a_{k}+4 k^{2} \nu a_{k +1}-8 k \,\mu ^{2} a_{k}+18 k \,\mu ^{2} a_{k +1}-8 k \,\nu ^{2} a_{k}+18 k \,\nu ^{2} a_{k +1}+2 a_{k} \mu ^{3}-2 a_{k} \nu \,\mu ^{2}-2 a_{k} \nu ^{2} \mu +2 a_{k} \nu ^{3}+23 k^{2} a_{k}-134 k^{2} a_{k +1}-8 k \mu a_{k}+18 k \mu a_{k +1}-8 k \nu a_{k}+18 k \nu a_{k +1}-7 a_{k} \mu ^{2}+20 \mu ^{2} a_{k +1}-2 a_{k} \nu \mu -7 a_{k} \nu ^{2}+20 \nu ^{2} a_{k +1}+28 k a_{k}-208 k a_{k +1}-8 a_{k} \mu +20 \mu a_{k +1}-8 a_{k} \nu +20 \nu a_{k +1}+12 a_{k}-120 a_{k +1}}{4 \left (k +3\right )^{3} \left (k +2\right )}, 32 a_{1}-6 a_{0} \left (-\mu ^{2}-\nu ^{2}-\mu -\nu +2\right )=0\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +1\right )^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} \left (x +1\right )^{k +1}\right ), a_{k +2}=-\frac {k^{4} a_{k}-4 k^{4} a_{k +1}-2 k^{2} \mu ^{2} a_{k}+4 k^{2} \mu ^{2} a_{k +1}-2 k^{2} \nu ^{2} a_{k}+4 k^{2} \nu ^{2} a_{k +1}+\mu ^{4} a_{k}-2 \mu ^{2} \nu ^{2} a_{k}+\nu ^{4} a_{k}+4 k^{3} a_{k}-22 k^{3} a_{k +1}-2 k^{2} \mu a_{k}+4 k^{2} \mu a_{k +1}-2 k^{2} \nu a_{k}+4 k^{2} \nu a_{k +1}-4 k \,\mu ^{2} a_{k}+10 k \,\mu ^{2} a_{k +1}-4 k \,\nu ^{2} a_{k}+10 k \,\nu ^{2} a_{k +1}+2 \mu ^{3} a_{k}-2 \mu ^{2} \nu a_{k}-2 \mu \,\nu ^{2} a_{k}+2 \nu ^{3} a_{k}+5 k^{2} a_{k}-44 k^{2} a_{k +1}-4 k \mu a_{k}+10 k \mu a_{k +1}-4 k \nu a_{k}+10 k \nu a_{k +1}-\mu ^{2} a_{k}+6 \mu ^{2} a_{k +1}-2 \mu \nu a_{k}-\nu ^{2} a_{k}+6 \nu ^{2} a_{k +1}+2 k a_{k}-38 k a_{k +1}-2 \mu a_{k}+6 \mu a_{k +1}-2 \nu a_{k}+6 \nu a_{k +1}-12 a_{k +1}}{4 \left (k +2\right )^{3} \left (k +1\right )}, 0=0, b_{k +2}=-\frac {k^{4} b_{k}-4 k^{4} b_{k +1}-2 k^{2} \mu ^{2} b_{k}+4 k^{2} \mu ^{2} b_{k +1}-2 k^{2} \nu ^{2} b_{k}+4 k^{2} \nu ^{2} b_{k +1}+\mu ^{4} b_{k}-2 \mu ^{2} \nu ^{2} b_{k}+\nu ^{4} b_{k}+8 k^{3} b_{k}-38 k^{3} b_{k +1}-2 k^{2} \mu b_{k}+4 k^{2} \mu b_{k +1}-2 k^{2} \nu b_{k}+4 k^{2} \nu b_{k +1}-8 k \,\mu ^{2} b_{k}+18 k \,\mu ^{2} b_{k +1}-8 k \,\nu ^{2} b_{k}+18 k \,\nu ^{2} b_{k +1}+2 \mu ^{3} b_{k}-2 \mu ^{2} \nu b_{k}-2 \mu \,\nu ^{2} b_{k}+2 \nu ^{3} b_{k}+23 k^{2} b_{k}-134 k^{2} b_{k +1}-8 k \mu b_{k}+18 k \mu b_{k +1}-8 k \nu b_{k}+18 k \nu b_{k +1}-7 \mu ^{2} b_{k}+20 \mu ^{2} b_{k +1}-2 \mu \nu b_{k}-7 \nu ^{2} b_{k}+20 \nu ^{2} b_{k +1}+28 k b_{k}-208 k b_{k +1}-8 \mu b_{k}+20 \mu b_{k +1}-8 \nu b_{k}+20 \nu b_{k +1}+12 b_{k}-120 b_{k +1}}{4 \left (k +3\right )^{3} \left (k +2\right )}, 32 b_{1}-6 b_{0} \left (-\mu ^{2}-\nu ^{2}-\mu -\nu +2\right )=0\right ] \end {array} \]
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying high order exact linear fully integrable trying to convert to a linear ODE with constant coefficients trying differential order: 4; missing the dependent variable Multiplying solutions by`, exp(Int(2*x/((x-1)*(x+1)), x))`Equation is the symmetric product of`, diff(diff(y(x), x), x)-(nu^2*x^2+nu 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 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: 0.5 (sec). Leaf size: 35
dsolve((x^2-1)^2*diff(diff(diff(diff(y(x),x),x),x),x)+10*x*(x^2-1)*diff(diff(diff(y(x),x),x),x)+(24*x^2-8-2*(mu*(mu+1)+nu*(nu+1))*(x^2-1))*diff(diff(y(x),x),x)-6*x*(mu*(mu+1)+nu*(nu+1)-2)*diff(y(x),x)+((mu*(mu+1)-nu*(nu+1))^2-2*mu*(mu+1)-2*nu*(nu+1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\operatorname {LegendreQ}\left (\mu , x\right ) c_{2} +c_{1} \operatorname {LegendreP}\left (\mu , x\right )\right ) \operatorname {LegendreP}\left (\nu , x\right )+\operatorname {LegendreQ}\left (\nu , x\right ) \left (\operatorname {LegendreQ}\left (\mu , x\right ) c_{4} +c_{3} \operatorname {LegendreP}\left (\mu , x\right )\right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(-2*\[Mu]*(1 + \[Mu]) - 2*\[Nu]*(1 + \[Nu]) + (\[Mu]*(1 + \[Mu]) - \[Nu]*(1 + \[Nu]))^2)*y[x] - 6*(-2 + \[Mu]*(1 +\[Mu]) + \[Nu]*(1 + \[Nu]))*x*y'[x] + (-8 + 24*x^3 - 2*(\[Mu]*(1 + \[Mu]) + \[Nu]*(1 + \[Nu]))*(-1 + x^2))*y''[x] + 10*x*(-1 + x^2)*Derivative[3][y][x] + (-1 + x^2)^2*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved