2.29.8 Problem 68

Internal problem ID [13729]
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 : 68
Date solved : Friday, December 19, 2025 at 11:20:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.29.8.1 ✓ Maple. Time used: 0.164 (sec). Leaf size: 158

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

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

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 
      A Liouvillian solution exists 
      Reducible group (found an exponential solution) 
      Group is reducible, not completely reducible 
   <- Kovacics algorithm successful 
<- Equivalence, under non-integer power transformations successful
 

2.29.8.2 ✗ Mathematica

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

Not solved

2.29.8.3 ✗ Sympy

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