54.7.65 problem 1673 (book 6.82)

Internal problem ID [12914]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1673 (book 6.82)
Date solved : Wednesday, October 01, 2025 at 02:46:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : solve: Cannot solve -a*(-y(x) + y(x)**n) + x**2*Derivative(y(x), (x, 2))