2.34.6 Problem 244

Internal problem ID [13904]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-8. Other equations.
Problem number : 244
Date solved : Friday, December 19, 2025 at 08:26:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.34.6.1 ✓ Maple. Time used: 0.132 (sec). Leaf size: 137

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

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

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
 

2.34.6.2 ✗ Mathematica

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

Not solved

2.34.6.3 ✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-a*y(x) - x**n*Derivative(y(x), (x, 2)) +
 

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', '2nd_power_series_regular')