55.29.5 problem 65

Internal problem ID [13838]
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 : 65
Date solved : Thursday, October 02, 2025 at 08:07:34 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 43

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

\[ y = c_1 \sin \left (\frac {x^{-n +1} \sqrt {b}}{n -1}\right )+c_2 \cos \left (\frac {x^{-n +1} \sqrt {b}}{n -1}\right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 52

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

\begin{align*} y(x)&\to c_1 \cos \left (\frac {\sqrt {b} x^{1-n}}{n-1}\right )+c_2 \sin \left (\frac {\sqrt {b} x^{1-n}}{1-n}\right ) \end{align*}

✗ Sympy

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

TypeError : invalid input: 1 - n