Internal problem ID [9844]
Internal file name [OUTPUT/8789_Monday_June_06_2022_05_29_21_AM_86531150/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1520.
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {2 \left (x -\operatorname {a1} \right ) \left (x -\operatorname {a2} \right ) \left (x -\operatorname {a3} \right ) y^{\prime \prime \prime }+\left (9 x^{2}-6 \left (\operatorname {a1} +\operatorname {a2} +\operatorname {a3} \right ) x +3 \operatorname {a1} \operatorname {a2} +3 \operatorname {a1} \operatorname {a3} +3 \operatorname {a2} \operatorname {a3} \right ) y^{\prime \prime }-2 \left (\left (n^{2}+n -3\right ) x +b \right ) y^{\prime }-n \left (n +1\right ) y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 \left (x -\mathit {a1} \right ) \left (x -\mathit {a2} \right ) \left (x -\mathit {a3} \right ) \left (\frac {d}{d x}y^{\prime \prime }\right )+\left (9 x^{2}-6 \left (\mathit {a1} +\mathit {a2} +\mathit {a3} \right ) x +3 \mathit {a1} \mathit {a2} +3 \mathit {a1} \mathit {a3} +3 \mathit {a2} \mathit {a3} \right ) \left (\frac {d}{d x}y^{\prime }\right )-2 \left (\left (n^{2}+n -3\right ) x +b \right ) y^{\prime }-n \left (n +1\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \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 {3 \left (\mathit {a1} \mathit {a2} +\mathit {a1} \mathit {a3} -2 \mathit {a1} x +\mathit {a2} \mathit {a3} -2 x \mathit {a2} -2 x \mathit {a3} +3 x^{2}\right )}{2 \left (-x +\mathit {a1} \right ) \left (-x +\mathit {a2} \right ) \left (-x +\mathit {a3} \right )}, P_{3}\left (x \right )=\frac {n^{2} x +n x +b -3 x}{\left (-x +\mathit {a1} \right ) \left (-x +\mathit {a2} \right ) \left (-x +\mathit {a3} \right )}, P_{4}\left (x \right )=\frac {n \left (n +1\right )}{2 \left (-x +\mathit {a1} \right ) \left (-x +\mathit {a2} \right ) \left (-x +\mathit {a3} \right )}\right ] \\ {} & \circ & \left (x -\mathit {a1} \right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =\mathit {a1} \\ {} & {} & \left (\left (x -\mathit {a1} \right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}\mathit {a1}}}}=\frac {3 \left (\mathit {a1}^{2}-\mathit {a1} \mathit {a2} -\mathit {a1} \mathit {a3} +\mathit {a2} \mathit {a3} \right )}{2 \left (-\mathit {a1} +\mathit {a3} \right ) \left (-\mathit {a1} +\mathit {a2} \right )} \\ {} & \circ & \left (x -\mathit {a1} \right )^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =\mathit {a1} \\ {} & {} & \left (\left (x -\mathit {a1} \right )^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}\mathit {a1}}}}=0 \\ {} & \circ & \left (x -\mathit {a1} \right )^{3}\cdot P_{4}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =\mathit {a1} \\ {} & {} & \left (\left (x -\mathit {a1} \right )^{3}\cdot P_{4}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}\mathit {a1}}}}=0 \\ {} & \circ & x =\mathit {a1} \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}=\mathit {a1} \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & 2 \left (-x +\mathit {a1} \right ) \left (-x +\mathit {a2} \right ) \left (-x +\mathit {a3} \right ) \left (\frac {d}{d x}y^{\prime \prime }\right )+\left (-3 \mathit {a1} \mathit {a2} -3 \mathit {a1} \mathit {a3} +6 \mathit {a1} x -3 \mathit {a2} \mathit {a3} +6 \mathit {a2} x +6 \mathit {a3} x -9 x^{2}\right ) \left (\frac {d}{d x}y^{\prime }\right )+\left (2 n^{2} x +2 n x +2 b -6 x \right ) y^{\prime }+n \left (n +1\right ) y=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u +\mathit {a1} \hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & \left (-2 \mathit {a1}^{2} u +2 \mathit {a1} \mathit {a2} u +2 \mathit {a1} \mathit {a3} u -4 \mathit {a1} \,u^{2}-2 \mathit {a2} \mathit {a3} u +2 \mathit {a2} \,u^{2}+2 \mathit {a3} \,u^{2}-2 u^{3}\right ) \left (\frac {d}{d u}\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (-3 \mathit {a1}^{2}+3 \mathit {a1} \mathit {a2} +3 \mathit {a1} \mathit {a3} -12 \mathit {a1} u -3 \mathit {a2} \mathit {a3} +6 \mathit {a2} u +6 \mathit {a3} u -9 u^{2}\right ) \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )+\left (2 n^{2} \mathit {a1} +2 n^{2} u +2 n \mathit {a1} +2 n u -6 \mathit {a1} +2 b -6 u \right ) \left (\frac {d}{d u}y \left (u \right )\right )+\left (n^{2}+n \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} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & -a_{0} \left (2 r -1\right ) r \left (-1+r \right ) \left (\mathit {a1} -\mathit {a3} \right ) \left (\mathit {a1} -\mathit {a2} \right ) u^{-2+r}+\left (-a_{1} \left (2 r +1\right ) \left (1+r \right ) r \left (\mathit {a1} -\mathit {a3} \right ) \left (\mathit {a1} -\mathit {a2} \right )-2 a_{0} r \left (-n^{2} \mathit {a1} +2 \mathit {a1} \,r^{2}-\mathit {a2} \,r^{2}-\mathit {a3} \,r^{2}-n \mathit {a1} +\mathit {a1} +\mathit {a2} +\mathit {a3} -b \right )\right ) u^{-1+r}+\left (\moverset {\infty }{\munderset {k =0}{\sum }}\left (-a_{k +2} \left (2 r +3+2 k \right ) \left (k +2+r \right ) \left (k +1+r \right ) \left (\mathit {a1} -\mathit {a3} \right ) \left (\mathit {a1} -\mathit {a2} \right )-2 a_{k +1} \left (k +1+r \right ) \left (2 \left (k +1\right )^{2} \mathit {a1} +4 \mathit {a1} \left (k +1\right ) r -n^{2} \mathit {a1} +2 \mathit {a1} \,r^{2}-\left (k +1\right )^{2} \mathit {a2} -2 \mathit {a2} \left (k +1\right ) r -\mathit {a2} \,r^{2}-\left (k +1\right )^{2} \mathit {a3} -2 \mathit {a3} \left (k +1\right ) r -\mathit {a3} \,r^{2}-n \mathit {a1} +\mathit {a1} +\mathit {a2} +\mathit {a3} -b \right )-a_{k} \left (2 k +2 r +1\right ) \left (r +1+n +k \right ) \left (r -n +k \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -\left (2 r -1\right ) r \left (-1+r \right ) \left (\mathit {a1} -\mathit {a3} \right ) \left (\mathit {a1} -\mathit {a2} \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, 1, \frac {1}{2}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & -2 a_{k +2} \left (r +\frac {3}{2}+k \right ) \left (k +1+r \right ) \left (k +2+r \right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \right )+2 a_{k +1} \left (\left (-2 \mathit {a1} +\mathit {a2} +\mathit {a3} \right ) k^{2}-4 \left (1+r \right ) \left (\mathit {a1} -\frac {\mathit {a2}}{2}-\frac {\mathit {a3}}{2}\right ) k +\left (-2 \mathit {a1} +\mathit {a2} +\mathit {a3} \right ) r^{2}+\left (-4 \mathit {a1} +2 \mathit {a2} +2 \mathit {a3} \right ) r +\left (n^{2}+n -3\right ) \mathit {a1} +b \right ) \left (k +1+r \right )-2 \left (k +r +\frac {1}{2}\right ) a_{k} \left (r -n +k \right ) \left (r +1+n +k \right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {4 \mathit {a1} \,k^{3} a_{k +1}+12 \mathit {a1} \,k^{2} r a_{k +1}-2 \mathit {a1} k \,n^{2} a_{k +1}+12 \mathit {a1} k \,r^{2} a_{k +1}-2 \mathit {a1} \,n^{2} r a_{k +1}+4 \mathit {a1} \,r^{3} a_{k +1}-2 \mathit {a2} \,k^{3} a_{k +1}-6 \mathit {a2} \,k^{2} r a_{k +1}-6 \mathit {a2} k \,r^{2} a_{k +1}-2 \mathit {a2} \,r^{3} a_{k +1}-2 \mathit {a3} \,k^{3} a_{k +1}-6 \mathit {a3} \,k^{2} r a_{k +1}-6 \mathit {a3} k \,r^{2} a_{k +1}-2 \mathit {a3} \,r^{3} a_{k +1}+12 \mathit {a1} \,k^{2} a_{k +1}-2 \mathit {a1} k n a_{k +1}+24 \mathit {a1} k r a_{k +1}-2 \mathit {a1} \,n^{2} a_{k +1}-2 \mathit {a1} n r a_{k +1}+12 \mathit {a1} \,r^{2} a_{k +1}-6 \mathit {a2} \,k^{2} a_{k +1}-12 \mathit {a2} k r a_{k +1}-6 \mathit {a2} \,r^{2} a_{k +1}-6 \mathit {a3} \,k^{2} a_{k +1}-12 \mathit {a3} k r a_{k +1}-6 \mathit {a3} \,r^{2} a_{k +1}+2 k^{3} a_{k}+6 k^{2} r a_{k}-2 k \,n^{2} a_{k}+6 k \,r^{2} a_{k}-2 n^{2} r a_{k}+2 r^{3} a_{k}+14 \mathit {a1} k a_{k +1}-2 \mathit {a1} n a_{k +1}+14 \mathit {a1} r a_{k +1}-4 \mathit {a2} k a_{k +1}-4 \mathit {a2} r a_{k +1}-4 \mathit {a3} k a_{k +1}-4 \mathit {a3} r a_{k +1}-2 b k a_{k +1}-2 b r a_{k +1}+3 k^{2} a_{k}-2 k n a_{k}+6 k r a_{k}-a_{k} n^{2}-2 n r a_{k}+3 r^{2} a_{k}+6 \mathit {a1} a_{k +1}-2 b a_{k +1}+a_{k} k -a_{k} n +r a_{k}}{\left (2 r +3+2 k \right ) \left (k +1+r \right ) \left (k +2+r \right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +2}=-\frac {4 \mathit {a1} \,k^{3} a_{k +1}-2 \mathit {a1} k \,n^{2} a_{k +1}-2 \mathit {a2} \,k^{3} a_{k +1}-2 \mathit {a3} \,k^{3} a_{k +1}+12 \mathit {a1} \,k^{2} a_{k +1}-2 \mathit {a1} k n a_{k +1}-2 \mathit {a1} \,n^{2} a_{k +1}-6 \mathit {a2} \,k^{2} a_{k +1}-6 \mathit {a3} \,k^{2} a_{k +1}+2 k^{3} a_{k}-2 k \,n^{2} a_{k}+14 \mathit {a1} k a_{k +1}-2 \mathit {a1} n a_{k +1}-4 \mathit {a2} k a_{k +1}-4 \mathit {a3} k a_{k +1}-2 b k a_{k +1}+3 k^{2} a_{k}-2 k n a_{k}-a_{k} n^{2}+6 \mathit {a1} a_{k +1}-2 b a_{k +1}+a_{k} k -a_{k} n}{\left (2 k +3\right ) \left (k +1\right ) \left (k +2\right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \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 {4 \mathit {a1} \,k^{3} a_{k +1}-2 \mathit {a1} k \,n^{2} a_{k +1}-2 \mathit {a2} \,k^{3} a_{k +1}-2 \mathit {a3} \,k^{3} a_{k +1}+12 \mathit {a1} \,k^{2} a_{k +1}-2 \mathit {a1} k n a_{k +1}-2 \mathit {a1} \,n^{2} a_{k +1}-6 \mathit {a2} \,k^{2} a_{k +1}-6 \mathit {a3} \,k^{2} a_{k +1}+2 k^{3} a_{k}-2 k \,n^{2} a_{k}+14 \mathit {a1} k a_{k +1}-2 \mathit {a1} n a_{k +1}-4 \mathit {a2} k a_{k +1}-4 \mathit {a3} k a_{k +1}-2 b k a_{k +1}+3 k^{2} a_{k}-2 k n a_{k}-a_{k} n^{2}+6 \mathit {a1} a_{k +1}-2 b a_{k +1}+a_{k} k -a_{k} n}{\left (2 k +3\right ) \left (k +1\right ) \left (k +2\right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \right )}, 0=0\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x -\mathit {a1} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x -\mathit {a1} \right )^{k}, a_{k +2}=-\frac {4 \mathit {a1} \,k^{3} a_{k +1}-2 \mathit {a1} k \,n^{2} a_{k +1}-2 \mathit {a2} \,k^{3} a_{k +1}-2 \mathit {a3} \,k^{3} a_{k +1}+12 \mathit {a1} \,k^{2} a_{k +1}-2 \mathit {a1} k n a_{k +1}-2 \mathit {a1} \,n^{2} a_{k +1}-6 \mathit {a2} \,k^{2} a_{k +1}-6 \mathit {a3} \,k^{2} a_{k +1}+2 k^{3} a_{k}-2 k \,n^{2} a_{k}+14 \mathit {a1} k a_{k +1}-2 \mathit {a1} n a_{k +1}-4 \mathit {a2} k a_{k +1}-4 \mathit {a3} k a_{k +1}-2 b k a_{k +1}+3 k^{2} a_{k}-2 k n a_{k}-a_{k} n^{2}+6 \mathit {a1} a_{k +1}-2 b a_{k +1}+a_{k} k -a_{k} n}{\left (2 k +3\right ) \left (k +1\right ) \left (k +2\right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \right )}, 0=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =1 \\ {} & {} & a_{k +2}=-\frac {4 \mathit {a1} \,k^{3} a_{k +1}-2 \mathit {a1} k \,n^{2} a_{k +1}-2 \mathit {a2} \,k^{3} a_{k +1}-2 \mathit {a3} \,k^{3} a_{k +1}+24 \mathit {a1} \,k^{2} a_{k +1}-2 \mathit {a1} k n a_{k +1}-4 \mathit {a1} \,n^{2} a_{k +1}-12 \mathit {a2} \,k^{2} a_{k +1}-12 \mathit {a3} \,k^{2} a_{k +1}+2 k^{3} a_{k}-2 k \,n^{2} a_{k}+50 \mathit {a1} k a_{k +1}-4 \mathit {a1} n a_{k +1}-22 \mathit {a2} k a_{k +1}-22 \mathit {a3} k a_{k +1}-2 b k a_{k +1}+9 k^{2} a_{k}-2 k n a_{k}-3 a_{k} n^{2}+36 \mathit {a1} a_{k +1}-12 \mathit {a2} a_{k +1}-12 \mathit {a3} a_{k +1}-4 b a_{k +1}+13 a_{k} k -3 a_{k} n +6 a_{k}}{\left (5+2 k \right ) \left (k +2\right ) \left (k +3\right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \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 {4 \mathit {a1} \,k^{3} a_{k +1}-2 \mathit {a1} k \,n^{2} a_{k +1}-2 \mathit {a2} \,k^{3} a_{k +1}-2 \mathit {a3} \,k^{3} a_{k +1}+24 \mathit {a1} \,k^{2} a_{k +1}-2 \mathit {a1} k n a_{k +1}-4 \mathit {a1} \,n^{2} a_{k +1}-12 \mathit {a2} \,k^{2} a_{k +1}-12 \mathit {a3} \,k^{2} a_{k +1}+2 k^{3} a_{k}-2 k \,n^{2} a_{k}+50 \mathit {a1} k a_{k +1}-4 \mathit {a1} n a_{k +1}-22 \mathit {a2} k a_{k +1}-22 \mathit {a3} k a_{k +1}-2 b k a_{k +1}+9 k^{2} a_{k}-2 k n a_{k}-3 a_{k} n^{2}+36 \mathit {a1} a_{k +1}-12 \mathit {a2} a_{k +1}-12 \mathit {a3} a_{k +1}-4 b a_{k +1}+13 a_{k} k -3 a_{k} n +6 a_{k}}{\left (5+2 k \right ) \left (k +2\right ) \left (k +3\right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \right )}, -6 a_{1} \left (\mathit {a1} -\mathit {a3} \right ) \left (\mathit {a1} -\mathit {a2} \right )-2 a_{0} \left (-n^{2} \mathit {a1} -n \mathit {a1} +3 \mathit {a1} -b \right )=0\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x -\mathit {a1} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x -\mathit {a1} \right )^{k +1}, a_{k +2}=-\frac {4 \mathit {a1} \,k^{3} a_{k +1}-2 \mathit {a1} k \,n^{2} a_{k +1}-2 \mathit {a2} \,k^{3} a_{k +1}-2 \mathit {a3} \,k^{3} a_{k +1}+24 \mathit {a1} \,k^{2} a_{k +1}-2 \mathit {a1} k n a_{k +1}-4 \mathit {a1} \,n^{2} a_{k +1}-12 \mathit {a2} \,k^{2} a_{k +1}-12 \mathit {a3} \,k^{2} a_{k +1}+2 k^{3} a_{k}-2 k \,n^{2} a_{k}+50 \mathit {a1} k a_{k +1}-4 \mathit {a1} n a_{k +1}-22 \mathit {a2} k a_{k +1}-22 \mathit {a3} k a_{k +1}-2 b k a_{k +1}+9 k^{2} a_{k}-2 k n a_{k}-3 a_{k} n^{2}+36 \mathit {a1} a_{k +1}-12 \mathit {a2} a_{k +1}-12 \mathit {a3} a_{k +1}-4 b a_{k +1}+13 a_{k} k -3 a_{k} n +6 a_{k}}{\left (5+2 k \right ) \left (k +2\right ) \left (k +3\right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \right )}, -6 a_{1} \left (\mathit {a1} -\mathit {a3} \right ) \left (\mathit {a1} -\mathit {a2} \right )-2 a_{0} \left (-n^{2} \mathit {a1} -n \mathit {a1} +3 \mathit {a1} -b \right )=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{2} \\ {} & {} & a_{k +2}=-\frac {-\frac {15}{4} \mathit {a2} a_{k +1}-\frac {15}{4} \mathit {a3} a_{k +1}+4 \mathit {a1} \,k^{3} a_{k +1}-2 \mathit {a2} \,k^{3} a_{k +1}-2 \mathit {a3} \,k^{3} a_{k +1}+18 \mathit {a1} \,k^{2} a_{k +1}-3 \mathit {a1} \,n^{2} a_{k +1}-9 \mathit {a2} \,k^{2} a_{k +1}-9 \mathit {a3} \,k^{2} a_{k +1}-2 k \,n^{2} a_{k}+29 \mathit {a1} k a_{k +1}-3 \mathit {a1} n a_{k +1}-\frac {23}{2} \mathit {a2} k a_{k +1}-\frac {23}{2} \mathit {a3} k a_{k +1}-2 b k a_{k +1}-2 k n a_{k}+\frac {3}{2} a_{k}-2 a_{k} n^{2}-2 a_{k} n -2 \mathit {a1} k \,n^{2} a_{k +1}-2 \mathit {a1} k n a_{k +1}+\frac {11}{2} a_{k} k +2 k^{3} a_{k}+6 k^{2} a_{k}+\frac {33}{2} \mathit {a1} a_{k +1}-3 b a_{k +1}}{\left (4+2 k \right ) \left (k +\frac {3}{2}\right ) \left (k +\frac {5}{2}\right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {1}{2} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +\frac {1}{2}}, a_{k +2}=-\frac {-\frac {15}{4} \mathit {a2} a_{k +1}-\frac {15}{4} \mathit {a3} a_{k +1}+4 \mathit {a1} \,k^{3} a_{k +1}-2 \mathit {a2} \,k^{3} a_{k +1}-2 \mathit {a3} \,k^{3} a_{k +1}+18 \mathit {a1} \,k^{2} a_{k +1}-3 \mathit {a1} \,n^{2} a_{k +1}-9 \mathit {a2} \,k^{2} a_{k +1}-9 \mathit {a3} \,k^{2} a_{k +1}-2 k \,n^{2} a_{k}+29 \mathit {a1} k a_{k +1}-3 \mathit {a1} n a_{k +1}-\frac {23}{2} \mathit {a2} k a_{k +1}-\frac {23}{2} \mathit {a3} k a_{k +1}-2 b k a_{k +1}-2 k n a_{k}+\frac {3}{2} a_{k}-2 a_{k} n^{2}-2 a_{k} n -2 \mathit {a1} k \,n^{2} a_{k +1}-2 \mathit {a1} k n a_{k +1}+\frac {11}{2} a_{k} k +2 k^{3} a_{k}+6 k^{2} a_{k}+\frac {33}{2} \mathit {a1} a_{k +1}-3 b a_{k +1}}{\left (4+2 k \right ) \left (k +\frac {3}{2}\right ) \left (k +\frac {5}{2}\right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \right )}, -\frac {3 a_{1} \left (\mathit {a1} -\mathit {a3} \right ) \left (\mathit {a1} -\mathit {a2} \right )}{2}-a_{0} \left (-n^{2} \mathit {a1} -n \mathit {a1} +\frac {3}{2} \mathit {a1} +\frac {3}{4} \mathit {a2} +\frac {3}{4} \mathit {a3} -b \right )=0\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x -\mathit {a1} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x -\mathit {a1} \right )^{k +\frac {1}{2}}, a_{k +2}=-\frac {-\frac {15}{4} \mathit {a2} a_{k +1}-\frac {15}{4} \mathit {a3} a_{k +1}+4 \mathit {a1} \,k^{3} a_{k +1}-2 \mathit {a2} \,k^{3} a_{k +1}-2 \mathit {a3} \,k^{3} a_{k +1}+18 \mathit {a1} \,k^{2} a_{k +1}-3 \mathit {a1} \,n^{2} a_{k +1}-9 \mathit {a2} \,k^{2} a_{k +1}-9 \mathit {a3} \,k^{2} a_{k +1}-2 k \,n^{2} a_{k}+29 \mathit {a1} k a_{k +1}-3 \mathit {a1} n a_{k +1}-\frac {23}{2} \mathit {a2} k a_{k +1}-\frac {23}{2} \mathit {a3} k a_{k +1}-2 b k a_{k +1}-2 k n a_{k}+\frac {3}{2} a_{k}-2 a_{k} n^{2}-2 a_{k} n -2 \mathit {a1} k \,n^{2} a_{k +1}-2 \mathit {a1} k n a_{k +1}+\frac {11}{2} a_{k} k +2 k^{3} a_{k}+6 k^{2} a_{k}+\frac {33}{2} \mathit {a1} a_{k +1}-3 b a_{k +1}}{\left (4+2 k \right ) \left (k +\frac {3}{2}\right ) \left (k +\frac {5}{2}\right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \right )}, -\frac {3 a_{1} \left (\mathit {a1} -\mathit {a3} \right ) \left (\mathit {a1} -\mathit {a2} \right )}{2}-a_{0} \left (-n^{2} \mathit {a1} -n \mathit {a1} +\frac {3}{2} \mathit {a1} +\frac {3}{4} \mathit {a2} +\frac {3}{4} \mathit {a3} -b \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 -\mathit {a1} \right )^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}c_{k} \left (x -\mathit {a1} \right )^{k +1}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}d_{k} \left (x -\mathit {a1} \right )^{k +\frac {1}{2}}\right ), a_{k +2}=-\frac {4 \mathit {a1} \,k^{3} a_{k +1}-2 \mathit {a1} k \,n^{2} a_{k +1}-2 \mathit {a2} \,k^{3} a_{k +1}-2 \mathit {a3} \,k^{3} a_{k +1}+12 \mathit {a1} \,k^{2} a_{k +1}-2 \mathit {a1} k n a_{k +1}-2 \mathit {a1} \,n^{2} a_{k +1}-6 \mathit {a2} \,k^{2} a_{k +1}-6 \mathit {a3} \,k^{2} a_{k +1}+2 k^{3} a_{k}-2 k \,n^{2} a_{k}+14 \mathit {a1} k a_{k +1}-2 \mathit {a1} n a_{k +1}-4 \mathit {a2} k a_{k +1}-4 \mathit {a3} k a_{k +1}-2 b k a_{k +1}+3 k^{2} a_{k}-2 k n a_{k}-n^{2} a_{k}+6 \mathit {a1} a_{k +1}-2 b a_{k +1}+k a_{k}-n a_{k}}{\left (2 k +3\right ) \left (k +1\right ) \left (k +2\right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \right )}, 0=0, c_{k +2}=-\frac {4 \mathit {a1} \,k^{3} c_{k +1}-2 \mathit {a1} k \,n^{2} c_{k +1}-2 \mathit {a2} \,k^{3} c_{k +1}-2 \mathit {a3} \,k^{3} c_{k +1}+24 \mathit {a1} \,k^{2} c_{k +1}-2 \mathit {a1} k n c_{k +1}-4 \mathit {a1} \,n^{2} c_{k +1}-12 \mathit {a2} \,k^{2} c_{k +1}-12 \mathit {a3} \,k^{2} c_{k +1}+2 k^{3} c_{k}-2 k \,n^{2} c_{k}+50 \mathit {a1} k c_{k +1}-4 \mathit {a1} n c_{k +1}-22 \mathit {a2} k c_{k +1}-22 \mathit {a3} k c_{k +1}-2 b k c_{k +1}+9 k^{2} c_{k}-2 k n c_{k}-3 n^{2} c_{k}+36 \mathit {a1} c_{k +1}-12 \mathit {a2} c_{k +1}-12 \mathit {a3} c_{k +1}-4 b c_{k +1}+13 k c_{k}-3 n c_{k}+6 c_{k}}{\left (5+2 k \right ) \left (k +2\right ) \left (k +3\right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \right )}, -6 c_{1} \left (\mathit {a1} -\mathit {a3} \right ) \left (\mathit {a1} -\mathit {a2} \right )-2 c_{0} \left (-n^{2} \mathit {a1} -n \mathit {a1} +3 \mathit {a1} -b \right )=0, d_{k +2}=-\frac {-2 \mathit {a1} k \,n^{2} d_{k +1}-2 \mathit {a1} k n d_{k +1}+\frac {3}{2} d_{k}-\frac {15}{4} \mathit {a2} d_{k +1}-\frac {15}{4} \mathit {a3} d_{k +1}-2 d_{k} n^{2}-2 d_{k} n +\frac {11}{2} d_{k} k +2 k^{3} d_{k}+6 k^{2} d_{k}+\frac {33}{2} \mathit {a1} d_{k +1}-3 b d_{k +1}+4 \mathit {a1} \,k^{3} d_{k +1}-2 \mathit {a2} \,k^{3} d_{k +1}-2 \mathit {a3} \,k^{3} d_{k +1}+18 \mathit {a1} \,k^{2} d_{k +1}-3 \mathit {a1} \,n^{2} d_{k +1}-9 \mathit {a2} \,k^{2} d_{k +1}-9 \mathit {a3} \,k^{2} d_{k +1}-2 k \,n^{2} d_{k}+29 \mathit {a1} k d_{k +1}-3 \mathit {a1} n d_{k +1}-\frac {23}{2} \mathit {a2} k d_{k +1}-\frac {23}{2} \mathit {a3} k d_{k +1}-2 b k d_{k +1}-2 k n d_{k}}{\left (4+2 k \right ) \left (k +\frac {3}{2}\right ) \left (k +\frac {5}{2}\right ) \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\mathit {a3} \right )}, -\frac {3 d_{1} \left (\mathit {a1} -\mathit {a3} \right ) \left (\mathit {a1} -\mathit {a2} \right )}{2}-d_{0} \left (-n^{2} \mathit {a1} -n \mathit {a1} +\frac {3}{2} \mathit {a1} +\frac {3}{4} \mathit {a2} +\frac {3}{4} \mathit {a3} -b \right )=0\right ] \end {array} \]
Maple trace
`Methods for third 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: 3; missing the dependent variable Equation is the 2nd symmetric power of diff(diff(y(x),x),x)-1/2*(a1*a2+a1*a3-2*a1*x+a2*a3-2*a2*x-2*x*a3+3*x^2)/(a1*a2*a3-a1*a2*x-a1* -> Attempting now to solve this lower order ODE 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 -> Kummer -> 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 HeunG ODE, case a <> 0, e <> 0, g <> 0, c = 0 `
✓ Solution by Maple
Time used: 0.812 (sec). Leaf size: 288
dsolve(2*(x-a1)*(x-a2)*(x-a3)*diff(diff(diff(y(x),x),x),x)+(9*x^2-6*(a1+a2+a3)*x+3*a1*a2+3*a1*a3+3*a2*a3)*diff(diff(y(x),x),x)-2*((n^2+n-3)*x+b)*diff(y(x),x)-n*(n+1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -c_{2} \left (x -\operatorname {a1} \right ) {\operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{-\operatorname {a2} +\operatorname {a1}}, \frac {\left (-n^{2}-n +3\right ) \operatorname {a1} -b}{-4 \operatorname {a2} +4 \operatorname {a1}}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right )}^{2}+c_{3} \operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{-\operatorname {a2} +\operatorname {a1}}, \frac {\left (-n^{2}-n +1\right ) \operatorname {a1} -b +\operatorname {a2} +\operatorname {a3}}{-4 \operatorname {a2} +4 \operatorname {a1}}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right ) \operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{-\operatorname {a2} +\operatorname {a1}}, \frac {\left (-n^{2}-n +3\right ) \operatorname {a1} -b}{-4 \operatorname {a2} +4 \operatorname {a1}}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right ) \sqrt {-x +\operatorname {a1}}+c_{1} {\operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{-\operatorname {a2} +\operatorname {a1}}, \frac {\left (-n^{2}-n +1\right ) \operatorname {a1} -b +\operatorname {a2} +\operatorname {a3}}{-4 \operatorname {a2} +4 \operatorname {a1}}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right )}^{2} \]
✓ Solution by Mathematica
Time used: 4.47 (sec). Leaf size: 418
DSolve[-(n*(1 + n)*y[x]) - 2*(b + (-3 + n + n^2)*x)*y'[x] + (3*a1*a2 + 3*a1*a3 + 3*a2*a3 - 6*(a1 + a2 + a3)*x + 9*x^2)*y''[x] + 2*(-a1 + x)*(-a2 + x)*(-a3 + x)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {c_3 (\text {a1}-x) \text {HeunG}\left [\frac {\text {a1}-\text {a3}}{\text {a1}-\text {a2}},-\frac {\text {a1} \left (n^2+n-3\right )+b}{4 (\text {a1}-\text {a2})},\frac {3}{4}-\frac {1}{4} \sqrt {(2 n+1)^2},\frac {1}{4} \left (\sqrt {(2 n+1)^2}+3\right ),\frac {3}{2},\frac {1}{2},\frac {\text {a1}-x}{\text {a1}-\text {a2}}\right ]^2}{\text {a1}-\text {a2}}+c_2 \sqrt {\frac {\text {a1}-x}{\text {a1}-\text {a2}}} \text {HeunG}\left [\frac {\text {a1}-\text {a3}}{\text {a1}-\text {a2}},\frac {-\text {a1} \left (n^2+n-1\right )+\text {a2}+\text {a3}-b}{4 (\text {a1}-\text {a2})},\frac {1}{4}-\frac {1}{4} \sqrt {(2 n+1)^2},\frac {1}{4} \left (\sqrt {(2 n+1)^2}+1\right ),\frac {1}{2},\frac {1}{2},\frac {\text {a1}-x}{\text {a1}-\text {a2}}\right ] \text {HeunG}\left [\frac {\text {a1}-\text {a3}}{\text {a1}-\text {a2}},-\frac {\text {a1} \left (n^2+n-3\right )+b}{4 (\text {a1}-\text {a2})},\frac {3}{4}-\frac {1}{4} \sqrt {(2 n+1)^2},\frac {1}{4} \left (\sqrt {(2 n+1)^2}+3\right ),\frac {3}{2},\frac {1}{2},\frac {\text {a1}-x}{\text {a1}-\text {a2}}\right ]+c_1 \text {HeunG}\left [\frac {\text {a1}-\text {a3}}{\text {a1}-\text {a2}},\frac {-\text {a1} \left (n^2+n-1\right )+\text {a2}+\text {a3}-b}{4 (\text {a1}-\text {a2})},\frac {1}{4}-\frac {1}{4} \sqrt {(2 n+1)^2},\frac {1}{4} \left (\sqrt {(2 n+1)^2}+1\right ),\frac {1}{2},\frac {1}{2},\frac {\text {a1}-x}{\text {a1}-\text {a2}}\right ]^2 \]