2.30.12 Problem 121

Internal problem ID [13782]
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 : 121
Date solved : Friday, December 19, 2025 at 12:25:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y&=0 \\ \end{align*}

2.30.12.1 ✓ Maple. Time used: 0.030 (sec). Leaf size: 177

ode:=x^2*diff(diff(y(x),x),x)+(a*x^(3*n)+b*x^(2*n)+1/4-1/4*n^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\frac {2 \,3^{{5}/{6}} \pi c_2 \left (a \,x^{n}+b \right ) \operatorname {BesselI}\left (\frac {1}{3}, \frac {2 \sqrt {\frac {-a^{3} x^{3 n}-3 a^{2} x^{2 n} b -3 a \,x^{n} b^{2}-b^{3}}{n^{2} a^{2}}}}{3}\right )}{3}+c_1 \operatorname {BesselI}\left (-\frac {1}{3}, \frac {2 \sqrt {\frac {-a^{3} x^{3 n}-3 a^{2} x^{2 n} b -3 a \,x^{n} b^{2}-b^{3}}{n^{2} a^{2}}}}{3}\right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{{2}/{3}} {\left (-\frac {\left (a \,x^{n}+b \right )^{3}}{n^{2} a^{2}}\right )}^{{1}/{3}}\right ) x^{\frac {1}{2}-\frac {n}{2}}}{3 {\left (-\frac {\left (a \,x^{n}+b \right )^{3}}{n^{2} a^{2}}\right )}^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )} \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying an equivalence, under non-integer power transformations, 
   to LODEs admitting Liouvillian solutions. 
   -> 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 
      <- hyper3 successful: indirect Equivalence to 0F1 under ``^ @ Moebius`` i\ 
s resolved 
   <- hypergeometric successful 
<- special function solution successful
 

2.30.12.2 ✗ Mathematica

ode=x^2*D[y[x],{x,2}]+(a*x^(3*n)+b*x^(2*n)+1/4-1/4*n^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.30.12.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
n = symbols("n") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (a*x**(3*n) + b*x**(2*n) - n**2/4 + 1/4)*y(x),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)) 
 
('2nd_power_series_regular',)