23.4.60 problem 60

Internal problem ID [6362]
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 : 60
Date solved : Tuesday, September 30, 2025 at 02:52:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=A \,x^{a} y^{b} {y^{\prime }}^{c} \end{align*}

✗ Maple

ode:=diff(diff(y(x),x),x) = A*x^a*y(x)^b*diff(y(x),x)^c; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE -(Derivative(y(x), (x, 2))/(A*x**a*y(x)**b))**(1/c) + Derivative