Internal problem ID [9657]
Internal file name [OUTPUT/8599_Monday_June_06_2022_04_21_59_AM_82023685/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1330.
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.
\[ \boxed {y^{\prime \prime }+\frac {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}+\frac {\left (\operatorname {DD} x +E \right ) y}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -\left (\frac {d}{d x}y^{\prime }\right ) \left (-x +a \right ) \left (-x +b \right ) \left (-x +c \right )+\left (A \,x^{2}+B x +C \right ) y^{\prime }+\left (\mathit {DD} x +E \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 (\mathit {DD} x +E \right ) y}{\left (-x +a \right ) \left (-x +b \right ) \left (-x +c \right )}+\frac {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (-x +a \right ) \left (-x +b \right ) \left (-x +c \right )} \\ \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 {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (-x +a \right ) \left (-x +b \right ) \left (-x +c \right )}-\frac {\left (\mathit {DD} x +E \right ) y}{\left (-x +a \right ) \left (-x +b \right ) \left (-x +c \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 {A \,x^{2}+B x +C}{\left (-x +a \right ) \left (-x +b \right ) \left (-x +c \right )}, P_{3}\left (x \right )=-\frac {\mathit {DD} x +E}{\left (-x +a \right ) \left (-x +b \right ) \left (-x +c \right )}\right ] \\ {} & \circ & \left (x -a \right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =a \\ {} & {} & \left (\left (x -a \right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}a}}}=\frac {A \,a^{2}+B a +C}{\left (-a +b \right ) \left (c -a \right )} \\ {} & \circ & \left (x -a \right )^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =a \\ {} & {} & \left (\left (x -a \right )^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}a}}}=0 \\ {} & \circ & x =a \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}=a \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) \left (-x +a \right ) \left (-x +b \right ) \left (-x +c \right )+\left (-A \,x^{2}-B x -C \right ) y^{\prime }+\left (-\mathit {DD} x -E \right ) y=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u +a \hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & \left (-a^{2} u +a b u +a c u -2 a \,u^{2}-b c u +b \,u^{2}+c \,u^{2}-u^{3}\right ) \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )+\left (-A \,a^{2}-2 A a u -A \,u^{2}-B a -B u -C \right ) \left (\frac {d}{d u}y \left (u \right )\right )+\left (-\mathit {DD} a -\mathit {DD} u -E \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..1 \\ {} & {} & 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..2 \\ {} & {} & 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 =1..3 \\ {} & {} & 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} \\ {} & {} & -a_{0} r \left (A \,a^{2}+a^{2} r -a b r -a c r +b c r +B a -a^{2}+b a +a c -b c +C \right ) u^{r -1}+\left (-a_{1} \left (1+r \right ) \left (A \,a^{2}+a^{2} r -a b r -a c r +b c r +B a +C \right )-a_{0} \left (2 A a r +2 a \,r^{2}-b \,r^{2}-c \,r^{2}+B r +\mathit {DD} a -2 a r +b r +c r +E \right )\right ) u^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (-a_{k +1} \left (k +1+r \right ) \left (A \,a^{2}+a^{2} \left (k +1\right )+a^{2} r -a b \left (k +1\right )-a b r -a c \left (k +1\right )-a c r +b c \left (k +1\right )+b c r +B a -a^{2}+b a +a c -b c +C \right )-a_{k} \left (2 A a k +2 A a r +2 a \,k^{2}+4 a k r +2 a \,r^{2}-b \,k^{2}-2 b k r -b \,r^{2}-c \,k^{2}-2 c k r -c \,r^{2}+B k +B r +\mathit {DD} a -2 a k -2 a r +b k +b r +c k +c r +E \right )-a_{k -1} \left (A \left (k -1\right )+A r +\left (k -1\right )^{2}+2 \left (k -1\right ) r +r^{2}+\mathit {DD} -k +1-r \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -r \left (A \,a^{2}+a^{2} r -a b r -a c r +b c r +B a -a^{2}+b a +a c -b c +C \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, -\frac {A \,a^{2}+B a -a^{2}+b a +a c -b c +C}{a^{2}-b a -a c +b c}\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & -a_{1} \left (1+r \right ) \left (A \,a^{2}+a^{2} r -a b r -a c r +b c r +B a +C \right )-a_{0} \left (2 A a r +2 a \,r^{2}-b \,r^{2}-c \,r^{2}+B r +\mathit {DD} a -2 a r +b r +c r +E \right )=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (-\left (-c +a \right ) \left (a -b \right ) a_{k +1}-2 a_{k} a +\left (b +c \right ) a_{k}-a_{k -1}\right ) k^{2}+\left (\left (-2 \left (-c +a \right ) \left (a -b \right ) a_{k +1}-4 a_{k} a +\left (2 b +2 c \right ) a_{k}-2 a_{k -1}\right ) r +\left (\left (-A -1\right ) a^{2}+\left (-B +b +c \right ) a -b c -C \right ) a_{k +1}-2 a_{k} \left (A -1\right ) a +\left (-B -b -c \right ) a_{k}-a_{k -1} \left (-3+A \right )\right ) k +\left (-\left (-c +a \right ) \left (a -b \right ) a_{k +1}-2 a_{k} a +\left (b +c \right ) a_{k}-a_{k -1}\right ) r^{2}+\left (\left (\left (-A -1\right ) a^{2}+\left (-B +b +c \right ) a -b c -C \right ) a_{k +1}-2 a_{k} \left (A -1\right ) a +\left (-B -b -c \right ) a_{k}-a_{k -1} \left (-3+A \right )\right ) r +\left (-A \,a^{2}-B a -C \right ) a_{k +1}-a_{k} \mathit {DD} a -a_{k} E +a_{k -1} \left (A -\mathit {DD} -2\right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & \left (-\left (-c +a \right ) \left (a -b \right ) a_{k +2}-2 a_{k +1} a +\left (b +c \right ) a_{k +1}-a_{k}\right ) \left (k +1\right )^{2}+\left (\left (-2 \left (-c +a \right ) \left (a -b \right ) a_{k +2}-4 a_{k +1} a +\left (2 b +2 c \right ) a_{k +1}-2 a_{k}\right ) r +\left (\left (-A -1\right ) a^{2}+\left (-B +b +c \right ) a -b c -C \right ) a_{k +2}-2 a_{k +1} \left (A -1\right ) a +\left (-B -b -c \right ) a_{k +1}-a_{k} \left (-3+A \right )\right ) \left (k +1\right )+\left (-\left (-c +a \right ) \left (a -b \right ) a_{k +2}-2 a_{k +1} a +\left (b +c \right ) a_{k +1}-a_{k}\right ) r^{2}+\left (\left (\left (-A -1\right ) a^{2}+\left (-B +b +c \right ) a -b c -C \right ) a_{k +2}-2 a_{k +1} \left (A -1\right ) a +\left (-B -b -c \right ) a_{k +1}-a_{k} \left (-3+A \right )\right ) r +\left (-A \,a^{2}-B a -C \right ) a_{k +2}-a_{k +1} \mathit {DD} a -a_{k +1} E +a_{k} \left (A -\mathit {DD} -2\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {2 A a k a_{k +1}+2 A a r a_{k +1}+2 a \,k^{2} a_{k +1}+4 a k r a_{k +1}+2 a \,r^{2} a_{k +1}-b \,k^{2} a_{k +1}-2 b k r a_{k +1}-b \,r^{2} a_{k +1}-c \,k^{2} a_{k +1}-2 c k r a_{k +1}-c \,r^{2} a_{k +1}+2 A a a_{k +1}+A k a_{k}+A r a_{k}+B k a_{k +1}+B r a_{k +1}+a_{k +1} \mathit {DD} a +2 a k a_{k +1}+2 a r a_{k +1}-b k a_{k +1}-b r a_{k +1}-c k a_{k +1}-c r a_{k +1}+k^{2} a_{k}+2 k r a_{k}+r^{2} a_{k}+B a_{k +1}+a_{k} \mathit {DD} +a_{k +1} E -a_{k} k -a_{k} r}{A \,a^{2} k +A \,a^{2} r +a^{2} k^{2}+2 a^{2} k r +a^{2} r^{2}-a b \,k^{2}-2 a b k r -a b \,r^{2}-a c \,k^{2}-2 a c k r -a c \,r^{2}+b c \,k^{2}+2 b c k r +b c \,r^{2}+2 A \,a^{2}+B a k +B a r +3 a^{2} k +3 a^{2} r -3 a b k -3 a b r -3 a c k -3 a c r +3 b c k +3 b c r +2 B a +C k +C r +2 a^{2}-2 b a -2 a c +2 b c +2 C} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +2}=-\frac {2 A a k a_{k +1}+2 a \,k^{2} a_{k +1}-b \,k^{2} a_{k +1}-c \,k^{2} a_{k +1}+2 A a a_{k +1}+A k a_{k}+B k a_{k +1}+a_{k +1} \mathit {DD} a +2 a k a_{k +1}-b k a_{k +1}-c k a_{k +1}+k^{2} a_{k}+B a_{k +1}+a_{k} \mathit {DD} +a_{k +1} E -a_{k} k}{A \,a^{2} k +a^{2} k^{2}-a b \,k^{2}-a c \,k^{2}+b c \,k^{2}+2 A \,a^{2}+B a k +3 a^{2} k -3 a b k -3 a c k +3 b c k +2 B a +C k +2 a^{2}-2 b a -2 a c +2 b c +2 C} \\ \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 {2 A a k a_{k +1}+2 a \,k^{2} a_{k +1}-b \,k^{2} a_{k +1}-c \,k^{2} a_{k +1}+2 A a a_{k +1}+A k a_{k}+B k a_{k +1}+a_{k +1} \mathit {DD} a +2 a k a_{k +1}-b k a_{k +1}-c k a_{k +1}+k^{2} a_{k}+B a_{k +1}+a_{k} \mathit {DD} +a_{k +1} E -a_{k} k}{A \,a^{2} k +a^{2} k^{2}-a b \,k^{2}-a c \,k^{2}+b c \,k^{2}+2 A \,a^{2}+B a k +3 a^{2} k -3 a b k -3 a c k +3 b c k +2 B a +C k +2 a^{2}-2 b a -2 a c +2 b c +2 C}, -a_{1} \left (A \,a^{2}+B a +C \right )-a_{0} \left (\mathit {DD} a +E \right )=0\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x -a \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x -a \right )^{k}, a_{k +2}=-\frac {2 A a k a_{k +1}+2 a \,k^{2} a_{k +1}-b \,k^{2} a_{k +1}-c \,k^{2} a_{k +1}+2 A a a_{k +1}+A k a_{k}+B k a_{k +1}+a_{k +1} \mathit {DD} a +2 a k a_{k +1}-b k a_{k +1}-c k a_{k +1}+k^{2} a_{k}+B a_{k +1}+a_{k} \mathit {DD} +a_{k +1} E -a_{k} k}{A \,a^{2} k +a^{2} k^{2}-a b \,k^{2}-a c \,k^{2}+b c \,k^{2}+2 A \,a^{2}+B a k +3 a^{2} k -3 a b k -3 a c k +3 b c k +2 B a +C k +2 a^{2}-2 b a -2 a c +2 b c +2 C}, -a_{1} \left (A \,a^{2}+B a +C \right )-a_{0} \left (\mathit {DD} a +E \right )=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {A \,a^{2}+B a -a^{2}+b a +a c -b c +C}{a^{2}-b a -a c +b c} \\ {} & {} & a_{k +2}=-\frac {-c \,k^{2} a_{k +1}+2 A a a_{k +1}+A k a_{k}+B k a_{k +1}+2 a k a_{k +1}-b k a_{k +1}-c k a_{k +1}+\frac {a_{k} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+a_{k +1} \mathit {DD} a -\frac {2 A a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {4 a k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}+\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} a_{k +1}}{\left (a^{2}-b a -a c +b c \right )^{2}}+\frac {2 b k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} a_{k +1}}{\left (a^{2}-b a -a c +b c \right )^{2}}+\frac {2 c k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} a_{k +1}}{\left (a^{2}-b a -a c +b c \right )^{2}}+a_{k} \mathit {DD} -a_{k} k +k^{2} a_{k}+B a_{k +1}+2 a \,k^{2} a_{k +1}-b \,k^{2} a_{k +1}-\frac {A \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k}}{a^{2}-b a -a c +b c}-\frac {B \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}+\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}+\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {2 k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k}}{a^{2}-b a -a c +b c}+\frac {\left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} a_{k}}{\left (a^{2}-b a -a c +b c \right )^{2}}+2 A a k a_{k +1}+a_{k +1} E}{2 C -2 b a +a^{2} k^{2}+3 a^{2} k +C k +2 A \,a^{2}+2 B a -2 a c -a c \,k^{2}+b c \,k^{2}+2 a^{2}+\frac {2 a b k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {a b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}+\frac {2 a c k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {a c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}+A \,a^{2} k -a b \,k^{2}-3 a c k +3 b c k +2 b c +B a k -3 a b k -\frac {2 b c k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {b c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {A \,a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {2 a^{2} k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {B a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {3 a b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {3 a c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {3 b c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {3 a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {C \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {A \,a^{2}+B a -a^{2}+b a +a c -b c +C}{a^{2}-b a -a c +b c} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k -\frac {A \,a^{2}+B a -a^{2}+b a +a c -b c +C}{a^{2}-b a -a c +b c}}, a_{k +2}=-\frac {-c \,k^{2} a_{k +1}+2 A a a_{k +1}+A k a_{k}+B k a_{k +1}+2 a k a_{k +1}-b k a_{k +1}-c k a_{k +1}+\frac {a_{k} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+a_{k +1} \mathit {DD} a -\frac {2 A a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {4 a k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}+\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} a_{k +1}}{\left (a^{2}-b a -a c +b c \right )^{2}}+\frac {2 b k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} a_{k +1}}{\left (a^{2}-b a -a c +b c \right )^{2}}+\frac {2 c k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} a_{k +1}}{\left (a^{2}-b a -a c +b c \right )^{2}}+a_{k} \mathit {DD} -a_{k} k +k^{2} a_{k}+B a_{k +1}+2 a \,k^{2} a_{k +1}-b \,k^{2} a_{k +1}-\frac {A \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k}}{a^{2}-b a -a c +b c}-\frac {B \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}+\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}+\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {2 k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k}}{a^{2}-b a -a c +b c}+\frac {\left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} a_{k}}{\left (a^{2}-b a -a c +b c \right )^{2}}+2 A a k a_{k +1}+a_{k +1} E}{2 C -2 b a +a^{2} k^{2}+3 a^{2} k +C k +2 A \,a^{2}+2 B a -2 a c -a c \,k^{2}+b c \,k^{2}+2 a^{2}+\frac {2 a b k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {a b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}+\frac {2 a c k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {a c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}+A \,a^{2} k -a b \,k^{2}-3 a c k +3 b c k +2 b c +B a k -3 a b k -\frac {2 b c k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {b c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {A \,a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {2 a^{2} k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {B a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {3 a b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {3 a c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {3 b c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {3 a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {C \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}}, -a_{1} \left (1-\frac {A \,a^{2}+B a -a^{2}+b a +a c -b c +C}{a^{2}-b a -a c +b c}\right ) \left (A \,a^{2}-\frac {a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {a b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {a c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {b c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+B a +C \right )-a_{0} \left (-\frac {2 A a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {B \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\mathit {DD} a +\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+E \right )=0\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x -a \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x -a \right )^{k -\frac {A \,a^{2}+B a -a^{2}+b a +a c -b c +C}{a^{2}-b a -a c +b c}}, a_{k +2}=-\frac {-c \,k^{2} a_{k +1}+2 A a a_{k +1}+A k a_{k}+B k a_{k +1}+2 a k a_{k +1}-b k a_{k +1}-c k a_{k +1}+\frac {a_{k} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+a_{k +1} \mathit {DD} a -\frac {2 A a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {4 a k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}+\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} a_{k +1}}{\left (a^{2}-b a -a c +b c \right )^{2}}+\frac {2 b k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} a_{k +1}}{\left (a^{2}-b a -a c +b c \right )^{2}}+\frac {2 c k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} a_{k +1}}{\left (a^{2}-b a -a c +b c \right )^{2}}+a_{k} \mathit {DD} -a_{k} k +k^{2} a_{k}+B a_{k +1}+2 a \,k^{2} a_{k +1}-b \,k^{2} a_{k +1}-\frac {A \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k}}{a^{2}-b a -a c +b c}-\frac {B \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}+\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}+\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k +1}}{a^{2}-b a -a c +b c}-\frac {2 k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) a_{k}}{a^{2}-b a -a c +b c}+\frac {\left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} a_{k}}{\left (a^{2}-b a -a c +b c \right )^{2}}+2 A a k a_{k +1}+a_{k +1} E}{2 C -2 b a +a^{2} k^{2}+3 a^{2} k +C k +2 A \,a^{2}+2 B a -2 a c -a c \,k^{2}+b c \,k^{2}+2 a^{2}+\frac {2 a b k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {a b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}+\frac {2 a c k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {a c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}+A \,a^{2} k -a b \,k^{2}-3 a c k +3 b c k +2 b c +B a k -3 a b k -\frac {2 b c k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {b c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {A \,a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {2 a^{2} k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {B a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {3 a b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {3 a c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {3 b c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {3 a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {C \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}}, -a_{1} \left (1-\frac {A \,a^{2}+B a -a^{2}+b a +a c -b c +C}{a^{2}-b a -a c +b c}\right ) \left (A \,a^{2}-\frac {a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {a b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {a c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {b c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+B a +C \right )-a_{0} \left (-\frac {2 A a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {B \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\mathit {DD} a +\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+E \right )=0\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}d_{k} \left (x -a \right )^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}e_{k} \left (x -a \right )^{k -\frac {A \,a^{2}+B a -a^{2}+b a +a c -b c +C}{a^{2}-b a -a c +b c}}\right ), d_{k +2}=-\frac {2 A a k d_{k +1}+2 a \,k^{2} d_{k +1}-b \,k^{2} d_{k +1}-c \,k^{2} d_{k +1}+2 A a d_{k +1}+A k d_{k}+B k d_{k +1}+\mathit {DD} a d_{k +1}+2 a k d_{k +1}-b k d_{k +1}-c k d_{k +1}+k^{2} d_{k}+B d_{k +1}+\mathit {DD} d_{k}+E d_{k +1}-k d_{k}}{A \,a^{2} k +a^{2} k^{2}-a b \,k^{2}-a c \,k^{2}+b c \,k^{2}+2 A \,a^{2}+B a k +3 a^{2} k -3 a b k -3 a c k +3 b c k +2 B a +C k +2 a^{2}-2 b a -2 a c +2 b c +2 C}, -d_{1} \left (A \,a^{2}+B a +C \right )-d_{0} \left (\mathit {DD} a +E \right )=0, e_{k +2}=-\frac {-c \,k^{2} e_{k +1}+2 A a e_{k +1}+A k e_{k}+B k e_{k +1}+2 a k e_{k +1}-b k e_{k +1}-c k e_{k +1}+\frac {e_{k} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+e_{k +1} \mathit {DD} a +2 a \,k^{2} e_{k +1}+e_{k} \mathit {DD} -e_{k} k +k^{2} e_{k}+B e_{k +1}+e_{k +1} E +\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} e_{k +1}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} e_{k +1}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} e_{k +1}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {A \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) e_{k}}{a^{2}-b a -a c +b c}-\frac {B \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) e_{k +1}}{a^{2}-b a -a c +b c}-\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) e_{k +1}}{a^{2}-b a -a c +b c}+\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) e_{k +1}}{a^{2}-b a -a c +b c}+\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) e_{k +1}}{a^{2}-b a -a c +b c}-\frac {2 k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) e_{k}}{a^{2}-b a -a c +b c}+2 A a k e_{k +1}-b \,k^{2} e_{k +1}+\frac {\left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2} e_{k}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {2 A a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) e_{k +1}}{a^{2}-b a -a c +b c}-\frac {4 a k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) e_{k +1}}{a^{2}-b a -a c +b c}+\frac {2 b k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) e_{k +1}}{a^{2}-b a -a c +b c}+\frac {2 c k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right ) e_{k +1}}{a^{2}-b a -a c +b c}}{2 C -2 b a +a^{2} k^{2}+3 a^{2} k +C k +2 A \,a^{2}+2 B a -2 a c +2 a^{2}-\frac {a b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {a c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}+2 b c -\frac {2 a^{2} k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-a c \,k^{2}+b c \,k^{2}+A \,a^{2} k -a b \,k^{2}-3 a c k +3 b c k +B a k -3 a b k +\frac {b c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}+\frac {2 a b k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {2 a c k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {2 b c k \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {A \,a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {B a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {3 a b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {3 a c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {3 b c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {3 a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {C \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}}, -e_{1} \left (1-\frac {A \,a^{2}+B a -a^{2}+b a +a c -b c +C}{a^{2}-b a -a c +b c}\right ) \left (A \,a^{2}-\frac {a^{2} \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {a b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {a c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {b c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+B a +C \right )-e_{0} \left (-\frac {2 A a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )^{2}}{\left (a^{2}-b a -a c +b c \right )^{2}}-\frac {B \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+\mathit {DD} a +\frac {2 a \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {b \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}-\frac {c \left (A \,a^{2}+B a -a^{2}+b a +a c -b c +C \right )}{a^{2}-b a -a c +b c}+E \right )=0\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 -> 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: 7.828 (sec). Leaf size: 1147
dsolve(diff(diff(y(x),x),x) = -(A*x^2+B*x+C)/(x-a)/(x-b)/(x-c)*diff(y(x),x)-(DD*x+E)/(x-a)/(x-b)/(x-c)*y(x),y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 10.917 (sec). Leaf size: 1166
DSolve[y''[x] == -(((E + DD*x)*y[x])/((-a + x)*(-b + x)*(-c + x))) - ((C + B*x + A*x^2)*y'[x])/((-a + x)*(-b + x)*(-c + x)),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (x-a)^{-\frac {C}{(a-b) (a-c)}} \left (c_2 \text {HeunG}\left [\frac {a-c}{a-b},\frac {A^2 b a^4+B^2 a^3+A \left (b^2-a b+(a+b) B+2 C\right ) a^3+(a-b)^2 (a \text {DD}+e) a^2-(a-b) (2 a-b) C a-B \left (a^3-b a^2-3 C a+b C\right ) a+(2 a-b) C^2+c \left (A^2 (a-2 b) a^3-b B^2 a+A \left (-a^3+(4 b+B) a^2-3 b (b+B) a-2 b C\right ) a-2 (a-b)^2 (a \text {DD}+e) a-C^2+(a+b) B \left (a^2-b a-C\right )+\left (3 a^2-4 b a+b^2\right ) C\right )+(a-b) c^2 \left ((A+\text {DD}) a^2+(-2 A b-\text {DD} b+e) a-C-b (B+e)\right )}{(a-b)^3 (a-c)^2},\frac {-\left ((A-1) a^2\right )-(A b+b+2 B+A c+c) a+A b c+b c-2 C+(a-b) (a-c) \sqrt {A^2-2 A-4 \text {DD}+1}}{2 (a-b) (a-c)},\frac {-\left ((b-c) \left (A^2-3 A-4 \text {DD}+2\right ) a^3\right )+\left (\left (A^2-A-4 \text {DD}+2\right ) b^2+2 B b+c \left (A^2-3 A-4 \text {DD}+2\right ) b-(A+1) B c-A C+C+c^2 \left (-3 A^2+5 A+8 \text {DD}-4\right )\right ) a^2+\left (-2 (A+2 \text {DD}-1) c^3+(-3 A B+B-2 b (A+2 \text {DD}-1)) c^2+\left (-2 \left (A^2-A-4 \text {DD}+2\right ) b^2-(A+3) B b-2 B^2-(A+3) C\right ) c-((A-1) b+2 B) C\right ) a-2 C^2+((A-1) b-2 B) c C+c^2 \left (\left (A^2-A-4 \text {DD}+2\right ) b^2+(A+1) B b-2 (A-1) C\right )+2 b c^3 (A+2 \text {DD}-1)+(a-c) \left (-\left ((A-2) (b-c) a^2\right )-\left ((A+2) b^2+2 B b-2 c^2-B c+C\right ) a+(A+2) b^2 c+b \left (-2 c^2+B c-C\right )+2 c C\right ) \sqrt {A^2-2 A-4 \text {DD}+1}}{2 (a-b) (a-c) \left ((A-1) c^2+(a+b+B) c-a b+C+(a-c) (b-c) \sqrt {A^2-2 A-4 \text {DD}+1}\right )},-\frac {(A-2) a^2+(2 b+B+2 c) a-2 b c+C}{(a-b) (a-c)},-\frac {A b^2+B b+C}{(a-b) (b-c)},\frac {a-x}{a-b}\right ] (x-a)^{-\frac {(A-1) a^2+(b+B+c) a-b c}{(a-b) (a-c)}}+c_1 \text {HeunG}\left [\frac {a-c}{a-b},\frac {a \text {DD}+e}{a-b},\frac {1}{2} \left (A+\sqrt {A^2-2 A-4 \text {DD}+1}-1\right ),\frac {4 \text {DD} c^2-B c+b \left (A^2-A-4 \text {DD}\right ) c-C+A (B c+C)-a (b-c) \left (A^2-A-4 \text {DD}\right )+(a A (b-c)-A b c-B c-C) \sqrt {A^2-2 A-4 \text {DD}+1}}{2 \left (A c^2-c^2+b c+B c+a (c-b)+C+(a-c) (b-c) \sqrt {A^2-2 A-4 \text {DD}+1}\right )},\frac {A a^2+B a+C}{(a-b) (a-c)},-\frac {A b^2+B b+C}{(a-b) (b-c)},\frac {a-x}{a-b}\right ] (x-a)^{\frac {C}{(a-b) (a-c)}}\right ) \]