2.31.31 Problem 179

Internal problem ID [13840]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-5
Problem number : 179
Date solved : Friday, December 19, 2025 at 03:01:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.31.31.1 ✓ Maple. Time used: 0.017 (sec). Leaf size: 501

ode:=(a__2*x^2+b__2*x+c__2)*diff(diff(y(x),x),x)+(b__1*x+c__1)*diff(y(x),x)+c__0*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_3 \operatorname {hypergeom}\left (\left [\frac {-a_{2} +b_{1} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, -\frac {a_{2} -b_{1} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}\right ], \left [\frac {b_{1} \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2} -2 a_{2} c_{1} +b_{1} b_{2}}{2 a_{2}^{2} \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}}\right ], \frac {\left (-2 x \,a_{2}^{2}-a_{2} b_{2} \right ) \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}+4 a_{2} c_{2} -b_{2}^{2}}{8 a_{2} c_{2} -2 b_{2}^{2}}\right )+c_4 {\left (2 \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, x \,a_{2}^{2}+\sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, b_{2} a_{2} -4 a_{2} c_{2} +b_{2}^{2}\right )}^{\frac {a_{2} \left (a_{2} -\frac {b_{1}}{2}\right ) \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}+a_{2} c_{1} -\frac {b_{1} b_{2}}{2}}{\sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}} \operatorname {hypergeom}\left (\left [\frac {a_{2} \left (a_{2} -\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}\right ) \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}+2 a_{2} c_{1} -b_{1} b_{2}}{2 \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}, \frac {a_{2} \left (a_{2} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}\right ) \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}+2 a_{2} c_{1} -b_{1} b_{2}}{2 \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}\right ], \left [\frac {4 a_{2}^{2} \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}-b_{1} \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2} +2 a_{2} c_{1} -b_{1} b_{2}}{2 a_{2}^{2} \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}}\right ], \frac {\left (-2 x \,a_{2}^{2}-a_{2} b_{2} \right ) \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}+4 a_{2} c_{2} -b_{2}^{2}}{8 a_{2} c_{2} -2 b_{2}^{2}}\right ) \]

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} \\ {} & {} & \left (a_{2} x^{2}+b_{2} x +c_{2} \right ) \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+\left (b_{1} x +c_{1} \right ) \left (\frac {d}{d x}y \left (x \right )\right )+c_{0} 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 {c_{0} y \left (x \right )}{a_{2} x^{2}+b_{2} x +c_{2}}-\frac {\left (b_{1} x +c_{1} \right ) \left (\frac {d}{d x}y \left (x \right )\right )}{a_{2} x^{2}+b_{2} x +c_{2}} \\ \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 +c_{1} \right ) \left (\frac {d}{d x}y \left (x \right )\right )}{a_{2} x^{2}+b_{2} x +c_{2}}+\frac {c_{0} y \left (x \right )}{a_{2} x^{2}+b_{2} x +c_{2}}=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 +c_{1}}{a_{2} x^{2}+b_{2} x +c_{2}}, P_{3}\left (x \right )=\frac {c_{0}}{a_{2} x^{2}+b_{2} x +c_{2}}\right ] \\ {} & \circ & \left (x -\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}}\right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}} \\ {} & {} & \left (\left (x -\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}}\right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}}}}}=0 \\ {} & \circ & {\left (x -\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}}\right )}^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}} \\ {} & {} & \left ({\left (x -\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}}\right )}^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}}}}}=0 \\ {} & \circ & x =\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}}\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}=\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}} \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & \left (a_{2} x^{2}+b_{2} x +c_{2} \right ) \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+\left (b_{1} x +c_{1} \right ) \left (\frac {d}{d x}y \left (x \right )\right )+c_{0} y \left (x \right )=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u +\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}}\hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & \left (a_{2} u^{2}+u \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\right ) \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (b_{1} u -\frac {b_{1} b_{2}}{2 a_{2}}+\frac {b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}}+c_{1} \right ) \left (\frac {d}{d u}y \left (u \right )\right )+c_{0} 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^{2}}{d u^{2}}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..2 \\ {} & {} & u^{m}\cdot \left (\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 ) u^{k +r -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & u^{m}\cdot \left (\frac {d^{2}}{d u^{2}}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} \\ {} & {} & \frac {a_{0} r \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} r -2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} +b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}+2 a_{2} c_{1} -b_{1} b_{2} \right ) u^{-1+r}}{2 a_{2}}+\left (\moverset {\infty }{\munderset {k =0}{\sum }}\left (\frac {a_{k +1} \left (k +1+r \right ) \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} \left (k +1\right )+2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} r -2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} +b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}+2 a_{2} c_{1} -b_{1} b_{2} \right )}{2 a_{2}}+a_{k} \left (a_{2} k^{2}+2 a_{2} k r +a_{2} r^{2}-a_{2} k -a_{2} r +b_{1} k +b_{1} r +c_{0} \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & \frac {r \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} r -2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} +b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}+2 a_{2} c_{1} -b_{1} b_{2} \right )}{2 a_{2}}=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, \frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \frac {2 a_{k +1} \left (k +1+r \right ) \left (\left (k +r \right ) a_{2} +\frac {b_{1}}{2}\right ) \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}+2 a_{k} \left (k +r \right ) \left (k +r -1\right ) a_{2}^{2}+\left (2 c_{1} \left (k +1+r \right ) a_{k +1}+2 a_{k} \left (b_{1} k +b_{1} r +c_{0} \right )\right ) a_{2} -b_{1} b_{2} a_{k +1} \left (k +1+r \right )}{2 a_{2}}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}+2 a_{2} k r +a_{2} r^{2}-a_{2} k -a_{2} r +b_{1} k +b_{1} r +c_{0} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k^{2}+4 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k r +2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} r^{2}+2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k +2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} r +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, b_{1} k +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, b_{1} r +2 a_{2} c_{1} k +2 a_{2} c_{1} r -b_{1} b_{2} k -b_{1} b_{2} r +b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}+2 a_{2} c_{1} -b_{1} b_{2}} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}-a_{2} k +b_{1} k +c_{0} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, b_{1} k +2 a_{2} c_{1} k -b_{1} b_{2} k +b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}+2 a_{2} c_{1} -b_{1} b_{2}} \\ \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 +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}-a_{2} k +b_{1} k +c_{0} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, b_{1} k +2 a_{2} c_{1} k -b_{1} b_{2} k +b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}+2 a_{2} c_{1} -b_{1} b_{2}}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x -\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}} \\ {} & {} & \left [y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} {\left (x -\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}}\right )}^{k}, a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}-a_{2} k +b_{1} k +c_{0} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, b_{1} k +2 a_{2} c_{1} k -b_{1} b_{2} k +b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}+2 a_{2} c_{1} -b_{1} b_{2}}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}} \\ {} & {} & a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}+\frac {k \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}+\frac {\left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )^{2}}{4 a_{2} \left (-4 a_{2} c_{2} +b_{2}^{2}\right )}-a_{2} k -\frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}+b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}+c_{0} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 k \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )+\frac {\left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )^{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}+2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k +2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{2 a_{2}}+2 a_{2} c_{1} k +\frac {c_{1} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}-b_{1} b_{2} k -\frac {b_{1} b_{2} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +\frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}}, a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}+\frac {k \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}+\frac {\left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )^{2}}{4 a_{2} \left (-4 a_{2} c_{2} +b_{2}^{2}\right )}-a_{2} k -\frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}+b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}+c_{0} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 k \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )+\frac {\left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )^{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}+2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k +2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{2 a_{2}}+2 a_{2} c_{1} k +\frac {c_{1} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}-b_{1} b_{2} k -\frac {b_{1} b_{2} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x -\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}} \\ {} & {} & \left [y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} {\left (x -\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}}\right )}^{k +\frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}}, a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}+\frac {k \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}+\frac {\left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )^{2}}{4 a_{2} \left (-4 a_{2} c_{2} +b_{2}^{2}\right )}-a_{2} k -\frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}+b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}+c_{0} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 k \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )+\frac {\left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )^{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}+2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k +2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{2 a_{2}}+2 a_{2} c_{1} k +\frac {c_{1} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}-b_{1} b_{2} k -\frac {b_{1} b_{2} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y \left (x \right )=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} {\left (x -\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}}\right )}^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} {\left (x -\frac {-b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 a_{2}}\right )}^{k +\frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}}\right ), a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}-a_{2} k +b_{1} k +c_{0} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, b_{1} k +2 a_{2} c_{1} k -b_{1} b_{2} k +b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}+2 a_{2} c_{1} -b_{1} b_{2}}, b_{k +1}=-\frac {2 a_{2} b_{k} \left (a_{2} k^{2}+\frac {k \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}+\frac {\left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )^{2}}{4 a_{2} \left (-4 a_{2} c_{2} +b_{2}^{2}\right )}-a_{2} k -\frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}+b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}+c_{0} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 k \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )+\frac {\left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )^{2}}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}+2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} k +2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{2 a_{2}}+2 a_{2} c_{1} k +\frac {c_{1} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}-b_{1} b_{2} k -\frac {b_{1} b_{2} \left (2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}-2 a_{2} c_{1} +b_{1} b_{2} \right )}{2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\, a_{2}}}\right ] \end {array} \]

2.31.31.2 ✓ Mathematica. Time used: 3.613 (sec). Leaf size: 498

ode=(a2*x^2+b2*x+c2)*D[y[x],{x,2}]+(b1*x+c1)*D[y[x],x]+c0*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \operatorname {Hypergeometric2F1}\left (-\frac {\text {a2}-\text {b1}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}}{2 \text {a2}},\frac {-\text {a2}+\text {b1}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}}{2 \text {a2}},\frac {\text {b1} \left (\text {b2}+\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}\right )-2 \text {a2} \text {c1}}{2 \text {a2} \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}},\frac {\text {b2}+2 \text {a2} x+\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}{2 \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right )-c_2 2^{\frac {\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {b1}}{2 \text {a2}}-\frac {\text {c1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}-1} \exp \left (-\frac {i \pi \left (\text {b1} \left (\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}+\text {b2}\right )-2 \text {a2} \text {c1}\right )}{2 \text {a2} \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right ) \left (\frac {\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}+2 \text {a2} x+\text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right )^{-\frac {\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {b1}}{2 \text {a2}}+\frac {\text {c1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+1} \operatorname {Hypergeometric2F1}\left (\frac {\frac {2 \text {c1} \text {a2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {a2}-\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}-\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}}{2 \text {a2}},\frac {\frac {2 \text {c1} \text {a2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {a2}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}-\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}}{2 \text {a2}},-\frac {\frac {\text {b2} \text {b1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {b1}+\text {a2} \left (-\frac {2 \text {c1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}-4\right )}{2 \text {a2}},\frac {\text {b2}+2 \text {a2} x+\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}{2 \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right ) \end{align*}

2.31.31.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a__2 = symbols("a__2") 
b__1 = symbols("b__1") 
b__2 = symbols("b__2") 
c__0 = symbols("c__0") 
c__1 = symbols("c__1") 
c__2 = symbols("c__2") 
y = Function("y") 
ode = Eq(c__0*y(x) + (b__1*x + c__1)*Derivative(y(x), x) + (a__2*x**2 + b__2*x + c__2)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', '2nd_power_series_ordinary')