23.4.93 problem 93

Internal problem ID [6395]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 93
Date solved : Tuesday, September 30, 2025 at 02:56:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

NotImplementedError : solve: Cannot solve a*(1 - y(x)**n)*y(x) + x**2*Derivative(y(x), (x, 2))