2.30.27 Problem 136

Internal problem ID [13797]
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
Problem number : 136
Date solved : Friday, December 19, 2025 at 12:43:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b \right ) y&=0 \\ \end{align*}

2.30.27.1 ✓ Maple. Time used: 0.040 (sec). Leaf size: 243

ode:=x^2*diff(diff(y(x),x),x)+(a*x^2+b)*diff(y(x),x)+c*((a-c)*x^2+b)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {x}\, \left ({\mathrm e}^{-x \left (a -c \right )} \operatorname {HeunD}\left (-4 \sqrt {-b \left (-2 c +a \right )}, -4 \sqrt {-b \left (-2 c +a \right )}-1+\left (-4 a +8 c \right ) b , 8 \sqrt {-b \left (-2 c +a \right )}, -4 \sqrt {-b \left (-2 c +a \right )}+1+\left (-8 c +4 a \right ) b , \frac {\sqrt {-b \left (-2 c +a \right )}\, x -b}{\sqrt {-b \left (-2 c +a \right )}\, x +b}\right ) c_2 +{\mathrm e}^{\frac {-x^{2} c +b}{x}} \operatorname {HeunD}\left (4 \sqrt {-b \left (-2 c +a \right )}, -4 \sqrt {-b \left (-2 c +a \right )}-1+\left (-4 a +8 c \right ) b , 8 \sqrt {-b \left (-2 c +a \right )}, -4 \sqrt {-b \left (-2 c +a \right )}+1+\left (-8 c +4 a \right ) b , \frac {\sqrt {-b \left (-2 c +a \right )}\, x -b}{\sqrt {-b \left (-2 c +a \right )}\, x +b}\right ) c_1 \right ) \]

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 
   A Liouvillian solution exists 
   Reducible group (found an exponential solution) 
   Group is reducible, not completely reducible 
   Solution has integrals. Trying a special function solution free of integrals\ 
... 
   -> 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 @ Mo\ 
ebius 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a \ 
power @ Moebius 
      <- Heun successful: received ODE is equivalent to the  HeunD  ODE, case  \ 
c = 0 
   <- Kovacics algorithm successful
 

2.30.27.2 ✓ Mathematica. Time used: 0.556 (sec). Leaf size: 44

ode=x^2*D[y[x],{x,2}]+(a*x^2+b)*D[y[x],x]+c*((a-c)*x^2+b)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-c x} \left (c_2 \int _1^xe^{\frac {b}{K[1]}-a K[1]+2 c K[1]}dK[1]+c_1\right ) \end{align*}

2.30.27.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(c*(b + x**2*(a - c))*y(x) + x**2*Derivative(y(x), (x, 2)) + (a*x**2 + b)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-a*c*x**2*y(x) - b*c*y(x) + c**2*x**2*y(x) - x**2*Derivative(y(x), (x, 2)))/(a*x**2 + b) cannot be solved by the factorable group method