23.4.72 problem 72

Internal problem ID [6374]
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 : 72
Date solved : Tuesday, September 30, 2025 at 02:53:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : 
No algorithms are implemented to solve equation _Dummy_37 - x**(n - 2)*f(y(x)