23.4.220 problem 220

Internal problem ID [6522]
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 : 220
Date solved : Tuesday, September 30, 2025 at 03:02:39 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} \left (-y+x y^{\prime }\right )^{2}+x^{2} y y^{\prime \prime }&=3 y^{2} \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 51

ode:=(-y(x)+x*diff(y(x),x))^2+x^2*y(x)*diff(diff(y(x),x),x) = 3*y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= -\frac {\sqrt {10}\, \sqrt {x \left (-c_2 \,x^{5}+c_1 \right )}}{5 x} \\ y &= \frac {\sqrt {10}\, \sqrt {x \left (-c_2 \,x^{5}+c_1 \right )}}{5 x} \\ \end{align*}

✓ Mathematica. Time used: 0.178 (sec). Leaf size: 23

ode=(-y[x] + x*D[y[x],x])^2 + x^2*y[x]*D[y[x],{x,2}] == 3*y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_2 \sqrt {x^5+c_1}}{\sqrt {x}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*y(x)*Derivative(y(x), (x, 2)) + (x*Derivative(y(x), x) - y(x))**2 - 3*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt((-x**2*Derivative(y(x), (x, 2)) + 3*