2.28.6 Problem 16

Internal problem ID [13677]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 16
Date solved : Friday, December 19, 2025 at 10:26:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

2.28.6.1 ✓ Maple. Time used: 0.001 (sec). Leaf size: 48

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

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

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
<- linear_1 successful
 

2.28.6.2 ✗ Mathematica

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

Not solved

2.28.6.3 ✗ 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*(a*x**n - b*x**(2*n) + n*x**(n - 1))*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : Mul object cannot be interpreted as an integer
 

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_ordinary')