23.4.13 problem 13

Internal problem ID [6315]
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 : 13
Date solved : Tuesday, September 30, 2025 at 02:45:49 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]

\begin{align*} a \,x^{r} y^{s}+y^{\prime \prime }&=0 \end{align*}

✗ Maple

ode:=a*x^r*y(x)^s+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=a*x^r*y[x]^s + 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") 
r = symbols("r") 
s = symbols("s") 
y = Function("y") 
ode = Eq(a*x**r*y(x)**s + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve a*x**r*y(x)**s + Derivative(y(x), (x, 2))