2.30.33 Problem 142

Internal problem ID [13803]
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 : 142
Date solved : Friday, December 19, 2025 at 12:54:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 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 \\ \end{align*}

2.30.33.1 ✓ Maple. Time used: 0.039 (sec). Leaf size: 232

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

\[ y = x^{-\frac {c}{2}+\frac {1}{2}} {\mathrm e}^{\frac {x \left (-a^{2} x -2 b a +2 A \right )}{2 a}} \left (x^{-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}} \operatorname {HeunB}\left (-\sqrt {c^{2}-2 c -4 d +1}, \frac {\sqrt {2}\, \left (-b a +2 A \right )}{a^{{3}/{2}}}, -c -\frac {2 A b}{a^{2}}+\frac {2 B}{a}-1+\frac {2 A^{2}}{a^{3}}, \frac {\sqrt {2}\, \left (-b c +2 C \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right ) c_2 +x^{\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}} \operatorname {HeunB}\left (\sqrt {c^{2}-2 c -4 d +1}, \frac {\sqrt {2}\, \left (-b a +2 A \right )}{a^{{3}/{2}}}, -c -\frac {2 A b}{a^{2}}+\frac {2 B}{a}-1+\frac {2 A^{2}}{a^{3}}, \frac {\sqrt {2}\, \left (-b c +2 C \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right ) c_1 \right ) \]

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 
<- 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 @ Moebi\ 
us 
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
 

2.30.33.2 ✗ Mathematica

ode=x^2*D[y[x],{x,2}]+x*(a*x^2+b*x+c)*D[y[x],x]+(A*x^3+B*x^2+C0*x+d)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.30.33.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
B = symbols("B") 
C = symbols("C") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(a*x**2 + b*x + c)*Derivative(y(x), x) + (A*x**3 + B*x**2 + C*x + d)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None