29.33 problem 142

Internal problem ID [10965]
Internal file name [OUTPUT/10222_Sunday_December_31_2023_11_10_18_AM_21141346/index.tex]

Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-4 Equation of form \(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 142.
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.

\[ \boxed {x^{2} y^{\prime \prime }+x \left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y=0} \]

29.33.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y^{\prime }\right )+x \left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \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 (A \,x^{3}+B \,x^{2}+C x +d \right ) y}{x^{2}}-\frac {\left (a \,x^{2}+b x +c \right ) y^{\prime }}{x} \\ \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 }}{x}+\frac {\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y}{x^{2}}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=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}{x}, P_{3}\left (x \right )=\frac {A \,x^{3}+B \,x^{2}+C x +d}{x^{2}}\right ] \\ {} & \circ & x \cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x \cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=c \\ {} & \circ & x^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=d \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y^{\prime }\right )+x \left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..3 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..3 \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{2}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (c r +r^{2}+d -r \right ) x^{r}+\left (\left (c r +r^{2}+c +d +r \right ) a_{1}+a_{0} \left (b r +C \right )\right ) x^{1+r}+\left (\left (c r +r^{2}+2 c +d +3 r +2\right ) a_{2}+a_{1} \left (b r +C +b \right )+a_{0} \left (a r +B \right )\right ) x^{2+r}+\left (\moverset {\infty }{\munderset {k =3}{\sum }}\left (a_{k} \left (c k +c r +k^{2}+2 k r +r^{2}+d -k -r \right )+a_{k -1} \left (b \left (k -1\right )+b r +C \right )+a_{k -2} \left (a \left (k -2\right )+a r +B \right )+A a_{k -3}\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & c r +r^{2}+d -r =0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}, -\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right \} \\ \bullet & {} & \textrm {The coefficients of each power of}\hspace {3pt} x \hspace {3pt}\textrm {must be 0}\hspace {3pt} \\ {} & {} & \left [\left (c r +r^{2}+c +d +r \right ) a_{1}+a_{0} \left (b r +C \right )=0, \left (c r +r^{2}+2 c +d +3 r +2\right ) a_{2}+a_{1} \left (b r +C +b \right )+a_{0} \left (a r +B \right )=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{1}=-\frac {a_{0} \left (b r +C \right )}{c r +r^{2}+c +d +r}, a_{2}=-\frac {a_{0} \left (a \,r^{2} c +a \,r^{3}-b^{2} r^{2}+B c r +B \,r^{2}-2 b r C +a r c +a r d +a \,r^{2}-b^{2} r +B c +d B +B r -C^{2}-C b \right )}{c^{2} r^{2}+2 c \,r^{3}+r^{4}+3 c^{2} r +2 c r d +7 c \,r^{2}+2 r^{2} d +4 r^{3}+2 c^{2}+3 c d +7 c r +d^{2}+4 d r +5 r^{2}+2 c +2 d +2 r}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k^{2}+\left (c +2 r -1\right ) k +r^{2}+\left (-1+c \right ) r +d \right ) a_{k}+\left (a a_{k -2}+b a_{k -1}\right ) k +\left (a a_{k -2}+b a_{k -1}\right ) r +\left (B -2 a \right ) a_{k -2}+\left (C -b \right ) a_{k -1}+A a_{k -3}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +3 \\ {} & {} & \left (\left (k +3\right )^{2}+\left (c +2 r -1\right ) \left (k +3\right )+r^{2}+\left (-1+c \right ) r +d \right ) a_{k +3}+\left (a a_{k +1}+b a_{k +2}\right ) \left (k +3\right )+\left (a a_{k +1}+b a_{k +2}\right ) r +\left (B -2 a \right ) a_{k +1}+\left (C -b \right ) a_{k +2}+A a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +3}=-\frac {a k a_{k +1}+a r a_{k +1}+b k a_{k +2}+b r a_{k +2}+A a_{k}+B a_{k +1}+C a_{k +2}+a a_{k +1}+2 b a_{k +2}}{c k +c r +k^{2}+2 k r +r^{2}+3 c +d +5 k +5 r +6} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2} \\ {} & {} & a_{k +3}=-\frac {a k a_{k +1}+a \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) a_{k +1}+b k a_{k +2}+b \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) a_{k +2}+A a_{k}+B a_{k +1}+C a_{k +2}+a a_{k +1}+2 b a_{k +2}}{c k +c \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+k^{2}+2 k \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+{\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}+\frac {c}{2}+d +5 k +\frac {17}{2}+\frac {5 \sqrt {c^{2}-2 c -4 d +1}}{2}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}}, a_{k +3}=-\frac {a k a_{k +1}+a \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) a_{k +1}+b k a_{k +2}+b \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) a_{k +2}+A a_{k}+B a_{k +1}+C a_{k +2}+a a_{k +1}+2 b a_{k +2}}{c k +c \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+k^{2}+2 k \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+{\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}+\frac {c}{2}+d +5 k +\frac {17}{2}+\frac {5 \sqrt {c^{2}-2 c -4 d +1}}{2}}, a_{1}=-\frac {a_{0} \left (b \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+C \right )}{c \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+{\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}+\frac {c}{2}+d +\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}}, a_{2}=-\frac {a_{0} \left (a {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2} c +a {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{3}-b^{2} {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}+B c \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+B {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}-2 b \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) C +a \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) c +a \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) d +a {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}-b^{2} \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+B c +d B +B \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )-C^{2}-C b \right )}{c^{2} {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}+2 c {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{3}+{\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{4}+3 c^{2} \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+2 c \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) d +7 c {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}+2 {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2} d +4 {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{3}+2 c^{2}+3 c d +7 c \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+d^{2}+4 d \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+5 {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}+c +2 d +1+\sqrt {c^{2}-2 c -4 d +1}}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2} \\ {} & {} & a_{k +3}=-\frac {a k a_{k +1}+a \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) a_{k +1}+b k a_{k +2}+b \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) a_{k +2}+A a_{k}+B a_{k +1}+C a_{k +2}+a a_{k +1}+2 b a_{k +2}}{c k +c \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+k^{2}+2 k \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+{\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}+\frac {c}{2}+d +5 k -\frac {5 \sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {17}{2}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}}, a_{k +3}=-\frac {a k a_{k +1}+a \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) a_{k +1}+b k a_{k +2}+b \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) a_{k +2}+A a_{k}+B a_{k +1}+C a_{k +2}+a a_{k +1}+2 b a_{k +2}}{c k +c \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+k^{2}+2 k \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+{\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}+\frac {c}{2}+d +5 k -\frac {5 \sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {17}{2}}, a_{1}=-\frac {a_{0} \left (b \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+C \right )}{c \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+{\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}+\frac {c}{2}+d -\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}}, a_{2}=-\frac {a_{0} \left (a {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2} c +a {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{3}-b^{2} {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}+B c \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+B {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}-2 b \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) C +a \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) c +a \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) d +a {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}-b^{2} \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+B c +d B +B \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )-C^{2}-C b \right )}{c^{2} {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}+2 c {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{3}+{\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{4}+3 c^{2} \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+2 c \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) d +7 c {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}+2 {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2} d +4 {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{3}+2 c^{2}+3 c d +7 c \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+d^{2}+4 d \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+5 {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}+c +2 d -\sqrt {c^{2}-2 c -4 d +1}+1}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}e_{k} x^{k -\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}f_{k} x^{k -\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}}\right ), e_{k +3}=-\frac {a k e_{k +1}+a \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) e_{k +1}+b k e_{k +2}+b \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) e_{k +2}+A e_{k}+B e_{k +1}+C e_{k +2}+a e_{k +1}+2 b e_{k +2}}{c k +c \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+k^{2}+2 k \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+{\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}+\frac {c}{2}+d +5 k +\frac {17}{2}+\frac {5 \sqrt {c^{2}-2 c -4 d +1}}{2}}, e_{1}=-\frac {e_{0} \left (b \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+C \right )}{c \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+{\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}+\frac {c}{2}+d +\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}}, e_{2}=-\frac {e_{0} \left (a {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2} c +a {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{3}-b^{2} {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}+B c \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+B {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}-2 b \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) C +a \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) c +a \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) d +a {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}-b^{2} \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+B c +d B +B \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )-C^{2}-C b \right )}{c^{2} {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}+2 c {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{3}+{\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{4}+3 c^{2} \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+2 c \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right ) d +7 c {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}+2 {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2} d +4 {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{3}+2 c^{2}+3 c d +7 c \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+d^{2}+4 d \left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )+5 {\left (-\frac {c}{2}+\frac {1}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}\right )}^{2}+c +2 d +1+\sqrt {c^{2}-2 c -4 d +1}}, f_{k +3}=-\frac {a k f_{k +1}+a \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) f_{k +1}+b k f_{k +2}+b \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) f_{k +2}+A f_{k}+B f_{k +1}+C f_{k +2}+a f_{k +1}+2 b f_{k +2}}{c k +c \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+k^{2}+2 k \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+{\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}+\frac {c}{2}+d +5 k -\frac {5 \sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {17}{2}}, f_{1}=-\frac {f_{0} \left (b \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+C \right )}{c \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+{\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}+\frac {c}{2}+d -\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}}, f_{2}=-\frac {f_{0} \left (a {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2} c +a {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{3}-b^{2} {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}+B c \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+B {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}-2 b \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) C +a \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) c +a \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) d +a {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}-b^{2} \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+B c +d B +B \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )-C^{2}-C b \right )}{c^{2} {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}+2 c {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{3}+{\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{4}+3 c^{2} \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+2 c \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right ) d +7 c {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}+2 {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2} d +4 {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{3}+2 c^{2}+3 c d +7 c \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+d^{2}+4 d \left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )+5 {\left (-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}\right )}^{2}+c +2 d -\sqrt {c^{2}-2 c -4 d +1}+1}\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  HeunB  ODE, case  c = 0 `
 

✓ Solution by Maple

Time used: 0.328 (sec). Leaf size: 232

dsolve(x^2*diff(y(x),x$2)+x*(a*x^2+b*x+c)*diff(y(x),x)+(A*x^3+B*x^2+C*x+d)*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = x^{-\frac {c}{2}+\frac {1}{2}} {\mathrm e}^{\frac {x \left (-a^{2} x -2 a b +2 A \right )}{2 a}} \left (c_{1} x^{\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}} \operatorname {HeunB}\left (\sqrt {c^{2}-2 c -4 d +1}, \frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{\frac {3}{2}}}, -c -\frac {2 A b}{a^{2}}+\frac {2 B}{a}-1+\frac {2 A^{2}}{a^{3}}, \frac {\sqrt {2}\, \left (-b c +2 C \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )+c_{2} x^{-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}} \operatorname {HeunB}\left (-\sqrt {c^{2}-2 c -4 d +1}, \frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{\frac {3}{2}}}, -c -\frac {2 A b}{a^{2}}+\frac {2 B}{a}-1+\frac {2 A^{2}}{a^{3}}, \frac {\sqrt {2}\, \left (-b c +2 C \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )\right ) \]

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[x^2*y''[x]+x*(a*x^2+b*x+c)*y'[x]+(A*x^3+B*x^2+C0*x+d)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

Not solved