2.30.11 Problem 120

Internal problem ID [13781]
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 : 120
Date solved : Friday, December 19, 2025 at 12:24:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.30.11.1 ✓ Maple. Time used: 0.020 (sec). Leaf size: 90

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

\[ y = \sqrt {x}\, x^{-\frac {n}{2}} \left (\operatorname {WhittakerW}\left (-\frac {i b}{2 n \sqrt {a}}, \frac {i \sqrt {-1+4 c}}{2 n}, \frac {2 i \sqrt {a}\, x^{n}}{n}\right ) c_2 +\operatorname {WhittakerM}\left (-\frac {i b}{2 n \sqrt {a}}, \frac {i \sqrt {-1+4 c}}{2 n}, \frac {2 i \sqrt {a}\, x^{n}}{n}\right ) c_1 \right ) \]

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 
      <- hyper3 successful: received ODE is equivalent to the 1F1 ODE 
   <- Whittaker successful 
<- special function solution successful
 

2.30.11.2 ✓ Mathematica. Time used: 0.133 (sec). Leaf size: 236

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

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

2.30.11.3 ✗ Sympy

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

ValueError : Expected Expr or iterable but got None
 

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',)