55.29.23 problem 83

Internal problem ID [13856]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 83
Date solved : Thursday, October 02, 2025 at 08:07:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (c -1\right ) \left (a x +b \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.138 (sec). Leaf size: 102

ode:=x*diff(diff(y(x),x),x)+(a*x^2+b*x+c)*diff(y(x),x)+(c-1)*(a*x+b)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 1.014 (sec). Leaf size: 49

ode=x*D[y[x],{x,2}]+(a*x^2+b*x+c)*D[y[x],x]+(c-1)*(a*x+b)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (c - 1)*(a*x + b)*y(x) + (a*x**2 + b*x + c)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None