23.4.227 problem 227

Internal problem ID [6529]
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 : 227
Date solved : Friday, October 03, 2025 at 02:09:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 2 x^{2} y y^{\prime \prime }&=-y^{2}+x^{2} {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 23

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

\begin{align*} y &= 0 \\ y &= \frac {x \left (2 c_{2} \ln \left (x \right )+c_{1} \right )^{2}}{4 c_{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.138 (sec). Leaf size: 17

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

\begin{align*} y(x)&\to c_2 x\\ y(x)&\to c_2 x \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - sqrt((2*x**2*Derivative(y(x), (x, 2)) + y(