23.4.104 problem 104

Internal problem ID [6406]
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 : 104
Date solved : Friday, October 03, 2025 at 02:05:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }&=f \left (\frac {x y^{\prime }}{y}\right ) y \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 37

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

\[ y = {\mathrm e}^{\int _{}^{\ln \left (x \right )}\operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}-\frac {1}{-\textit {\_a} +\textit {\_a}^{2}-f \left (\textit {\_a} \right )}d \textit {\_a} -\textit {\_b} +c_1 \right )d \textit {\_b} +c_2} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : 
No algorithms are implemented to solve equation _Dummy_37*x**2 - f(_X0*x/y(x)