2.32.5 Problem 186

Internal problem ID [13847]
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 : 186
Date solved : Friday, December 19, 2025 at 03:26:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (c x +d \right ) y&=0 \\ \end{align*}

2.32.5.1 ✓ Maple. Time used: 0.043 (sec). Leaf size: 132

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

\[ y = x^{\frac {1}{2}-\frac {a}{2}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}} \left (\operatorname {KummerU}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}\, b +\left (-1+a \right ) b -2 d}{2 b}, 1+\sqrt {a^{2}-2 a -4 c +1}, \frac {b}{x}\right ) c_2 +\operatorname {KummerM}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}\, b +\left (-1+a \right ) b -2 d}{2 b}, 1+\sqrt {a^{2}-2 a -4 c +1}, \frac {b}{x}\right ) c_1 \right ) \]

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 
   -> Kummer 
      -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      <- hyper3 successful: received ODE is equivalent to the 1F1 ODE 
   <- Kummer successful 
<- special function solution successful
 

2.32.5.2 ✓ Mathematica. Time used: 0.228 (sec). Leaf size: 255

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

\begin{align*} y(x)&\to -i^{-\sqrt {a^2-2 a-4 c+1}+a+1} b^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (c_2 i^{2 \sqrt {a^2-2 a-4 c+1}} b^{\sqrt {a^2-2 a-4 c+1}} \left (\frac {1}{x}\right )^{\sqrt {a^2-2 a-4 c+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (a-\frac {2 d}{b}+\sqrt {a^2-2 a-4 c+1}-1\right ),\sqrt {a^2-2 a-4 c+1}+1,\frac {b}{x}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (a-\frac {2 d}{b}-\sqrt {a^2-2 a-4 c+1}-1\right ),1-\sqrt {a^2-2 a-4 c+1},\frac {b}{x}\right )\right ) \end{align*}

2.32.5.3 ✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-c*x*y(x) - d*y(x) - x**3*Derivative(y(x)
 

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',)