2.32.13 Problem 194

Internal problem ID [13855]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 194
Date solved : Friday, December 19, 2025 at 04:04:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x +a_{2} \right ) y^{\prime \prime }+x \left (b_{1} x +a_{1} \right ) y^{\prime }+\left (b_{0} x +a_{0} \right ) y&=0 \\ \end{align*}

2.32.13.1 ✓ Maple. Time used: 0.015 (sec). Leaf size: 319

ode:=x^2*(x+a__2)*diff(diff(y(x),x),x)+x*(b__1*x+a__1)*diff(y(x),x)+(b__0*x+a__0)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,x^{\frac {a_{2} -a_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{2 a_{2}}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2} -b_{1} a_{2} -\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+a_{1}}{2 a_{2}}, \frac {b_{1} a_{2} -a_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2}}{2 a_{2}}\right ], \left [\frac {a_{2} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{a_{2}}\right ], -\frac {x}{a_{2}}\right )+c_2 \,x^{-\frac {-a_{2} +a_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{2 a_{2}}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2} -b_{1} a_{2} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+a_{1}}{2 a_{2}}, \frac {b_{1} a_{2} -a_{1} -\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2}}{2 a_{2}}\right ], \left [\frac {a_{2} -\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{a_{2}}\right ], -\frac {x}{a_{2}}\right ) \]

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 
      <- heuristic approach successful 
   <- hypergeometric successful 
<- special function solution successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (x +a_{2} \right ) \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+x \left (b_{1} x +a_{1} \right ) \left (\frac {d}{d x}y \left (x \right )\right )+\left (b_{0} x +a_{0} \right ) y \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=-\frac {\left (b_{0} x +a_{0} \right ) y \left (x \right )}{x^{2} \left (x +a_{2} \right )}-\frac {\left (b_{1} x +a_{1} \right ) \left (\frac {d}{d x}y \left (x \right )\right )}{x \left (x +a_{2} \right )} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y \left (x \right )\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^{2}}{d x^{2}}y \left (x \right )+\frac {\left (b_{1} x +a_{1} \right ) \left (\frac {d}{d x}y \left (x \right )\right )}{x \left (x +a_{2} \right )}+\frac {\left (b_{0} x +a_{0} \right ) y \left (x \right )}{x^{2} \left (x +a_{2} \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 {b_{1} x +a_{1}}{x \left (x +a_{2} \right )}, P_{3}\left (x \right )=\frac {b_{0} x +a_{0}}{x^{2} \left (x +a_{2} \right )}\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}}}=\frac {a_{1}}{a_{2}} \\ {} & \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}}}=\frac {a_{0}}{a_{2}} \\ {} & \circ & x =0\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}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & x^{2} \left (x +a_{2} \right ) \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+x \left (b_{1} x +a_{1} \right ) \left (\frac {d}{d x}y \left (x \right )\right )+\left (b_{0} x +a_{0} \right ) y \left (x \right )=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \left (x \right ) \\ {} & {} & y \left (x \right )=\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 \left (x \right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & x^{m}\cdot y \left (x \right )=\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 \left (x \right )=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot \left (\frac {d}{d x}y \left (x \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..2 \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y \left (x \right )\right )=\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 \left (\frac {d}{d x}y \left (x \right )\right )=\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^{m}\cdot \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =2..3 \\ {} & {} & x^{m}\cdot \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & x^{m}\cdot \left (\frac {d^{2}}{d x^{2}}y \left (x \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 ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (a_{2} r^{2}+a_{1} r -a_{2} r +a_{0} \right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a_{k} \left (a_{2} k^{2}+2 a_{2} k r +a_{2} r^{2}+a_{1} k +a_{1} r -a_{2} k -a_{2} r +a_{0} \right )+a_{k -1} \left (b_{1} \left (k -1\right )+b_{1} r +\left (k -1\right )^{2}+2 \left (k -1\right ) r +r^{2}+b_{0} -k +1-r \right )\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & a_{2} r^{2}+a_{1} r -a_{2} r +a_{0} =0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-\frac {-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}, \frac {a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k^{2}+\left (2 r +b_{1} -3\right ) k +r^{2}+\left (b_{1} -3\right ) r +b_{0} -b_{1} +2\right ) a_{k -1}+a_{k} \left (a_{2} k^{2}+\left (2 a_{2} r +a_{1} -a_{2} \right ) k +a_{2} r^{2}+\left (a_{1} -a_{2} \right ) r +a_{0} \right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & \left (\left (k +1\right )^{2}+\left (2 r +b_{1} -3\right ) \left (k +1\right )+r^{2}+\left (b_{1} -3\right ) r +b_{0} -b_{1} +2\right ) a_{k}+a_{k +1} \left (a_{2} \left (k +1\right )^{2}+\left (2 a_{2} r +a_{1} -a_{2} \right ) \left (k +1\right )+a_{2} r^{2}+\left (a_{1} -a_{2} \right ) r +a_{0} \right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {\left (b_{1} k +b_{1} r +k^{2}+2 k r +r^{2}+b_{0} -k -r \right ) a_{k}}{a_{2} k^{2}+2 a_{2} k r +a_{2} r^{2}+a_{1} k +a_{1} r +a_{2} k +a_{2} r +a_{0} +a_{1}} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}} \\ {} & {} & a_{k +1}=-\frac {\left (b_{1} k -\frac {b_{1} \left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{2 a_{2}}+k^{2}-\frac {k \left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{a_{2}}+\frac {\left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )^{2}}{4 a_{2}^{2}}+b_{0} -k +\frac {-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}\right ) a_{k}}{a_{2} k^{2}-k \left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )+\frac {\left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )^{2}}{4 a_{2}}+a_{1} k -\frac {a_{1} \left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{2 a_{2}}+a_{2} k +\frac {a_{2}}{2}+\frac {a_{1}}{2}-\frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2}+a_{0}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}} \\ {} & {} & \left [y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\frac {-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}}, a_{k +1}=-\frac {\left (b_{1} k -\frac {b_{1} \left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{2 a_{2}}+k^{2}-\frac {k \left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{a_{2}}+\frac {\left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )^{2}}{4 a_{2}^{2}}+b_{0} -k +\frac {-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}\right ) a_{k}}{a_{2} k^{2}-k \left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )+\frac {\left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )^{2}}{4 a_{2}}+a_{1} k -\frac {a_{1} \left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{2 a_{2}}+a_{2} k +\frac {a_{2}}{2}+\frac {a_{1}}{2}-\frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2}+a_{0}}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}} \\ {} & {} & a_{k +1}=-\frac {\left (b_{1} k +\frac {b_{1} \left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{2 a_{2}}+k^{2}+\frac {k \left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{a_{2}}+\frac {\left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )^{2}}{4 a_{2}^{2}}+b_{0} -k -\frac {a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}\right ) a_{k}}{a_{2} k^{2}+k \left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )+\frac {\left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )^{2}}{4 a_{2}}+a_{1} k +\frac {a_{1} \left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{2 a_{2}}+a_{2} k +\frac {a_{2}}{2}+\frac {a_{1}}{2}+\frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2}+a_{0}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}} \\ {} & {} & \left [y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}}, a_{k +1}=-\frac {\left (b_{1} k +\frac {b_{1} \left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{2 a_{2}}+k^{2}+\frac {k \left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{a_{2}}+\frac {\left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )^{2}}{4 a_{2}^{2}}+b_{0} -k -\frac {a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}\right ) a_{k}}{a_{2} k^{2}+k \left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )+\frac {\left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )^{2}}{4 a_{2}}+a_{1} k +\frac {a_{1} \left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{2 a_{2}}+a_{2} k +\frac {a_{2}}{2}+\frac {a_{1}}{2}+\frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2}+a_{0}}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y \left (x \right )=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\frac {-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k +\frac {a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}}\right ), a_{k +1}=-\frac {\left (b_{1} k -\frac {b_{1} \left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{2 a_{2}}+k^{2}-\frac {k \left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{a_{2}}+\frac {\left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )^{2}}{4 a_{2}^{2}}+b_{0} -k +\frac {-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}\right ) a_{k}}{a_{2} k^{2}-k \left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )+\frac {\left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )^{2}}{4 a_{2}}+a_{1} k -\frac {a_{1} \left (-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{2 a_{2}}+a_{2} k +\frac {a_{2}}{2}+\frac {a_{1}}{2}-\frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2}+a_{0}}, b_{k +1}=-\frac {\left (b_{1} k +\frac {b_{1} \left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{2 a_{2}}+k^{2}+\frac {k \left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{a_{2}}+\frac {\left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )^{2}}{4 a_{2}^{2}}+b_{0} -k -\frac {a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}\right ) b_{k}}{a_{2} k^{2}+k \left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )+\frac {\left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )^{2}}{4 a_{2}}+a_{1} k +\frac {a_{1} \left (a_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}\right )}{2 a_{2}}+a_{2} k +\frac {a_{2}}{2}+\frac {a_{1}}{2}+\frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2}+a_{0}}\right ] \end {array} \]

2.32.13.2 ✓ Mathematica. Time used: 0.449 (sec). Leaf size: 384

ode=x^2*(x+a2)*D[y[x],{x,2}]+x*(b1*x+a1)*D[y[x],x]+(b0*x+a0)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {a2}^{-\frac {\sqrt {-4 \text {a0} \text {a2}+\text {a1}^2-2 \text {a1} \text {a2}+\text {a2}^2}-\text {a1}+\text {a2}}{2 \text {a2}}} x^{-\frac {\sqrt {-4 \text {a0} \text {a2}+\text {a1}^2-2 \text {a1} \text {a2}+\text {a2}^2}+\text {a1}-\text {a2}}{2 \text {a2}}} \left (c_2 x^{\frac {\sqrt {-4 \text {a0} \text {a2}+\text {a1}^2-2 \text {a1} \text {a2}+\text {a2}^2}}{\text {a2}}} \operatorname {Hypergeometric2F1}\left (\frac {-\text {a1}+\text {a2} \text {b1}+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2} (\text {a2}-4 \text {a0})}-\text {a2} \sqrt {(\text {b1}-1)^2-4 \text {b0}}}{2 \text {a2}},\frac {-\text {a1}+\text {a2} \text {b1}+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2} (\text {a2}-4 \text {a0})}+\text {a2} \sqrt {(\text {b1}-1)^2-4 \text {b0}}}{2 \text {a2}},\frac {\text {a2}+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2}^2-4 \text {a0} \text {a2}}}{\text {a2}},-\frac {x}{\text {a2}}\right )+c_1 \text {a2}^{\frac {\sqrt {-4 \text {a0} \text {a2}+\text {a1}^2-2 \text {a1} \text {a2}+\text {a2}^2}}{\text {a2}}} \operatorname {Hypergeometric2F1}\left (-\frac {\text {a1}-\text {a2} \text {b1}+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2} (\text {a2}-4 \text {a0})}+\text {a2} \sqrt {(\text {b1}-1)^2-4 \text {b0}}}{2 \text {a2}},-\frac {\text {a1}-\text {a2} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {b0}}\right )+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2} (\text {a2}-4 \text {a0})}}{2 \text {a2}},1-\frac {\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2}^2-4 \text {a0} \text {a2}}}{\text {a2}},-\frac {x}{\text {a2}}\right )\right ) \end{align*}

2.32.13.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a__0 = symbols("a__0") 
a__1 = symbols("a__1") 
a__2 = symbols("a__2") 
b__0 = symbols("b__0") 
b__1 = symbols("b__1") 
y = Function("y") 
ode = Eq(x**2*(a__2 + x)*Derivative(y(x), (x, 2)) + x*(a__1 + b__1*x)*Derivative(y(x), x) + (a__0 + b__0*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None