54.3.392 problem 1409

Internal problem ID [12687]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1409
Date solved : Wednesday, October 01, 2025 at 02:20:04 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 44

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

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

✗ Sympy

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

TypeError : invalid input: 1 - a