23.4.225 problem 225

Internal problem ID [6527]
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 : 225
Date solved : Friday, October 03, 2025 at 02:09:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} \left (-y+x y^{\prime }\right )^{2}+x^{2} \left (x -y\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 25

ode:=(-y(x)+x*diff(y(x),x))^2+x^2*(x-y(x))*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= x \\ y &= \frac {x \left (c_2 +{\mathrm e}^{\frac {-x +c_1}{x}}\right )}{c_2} \\ \end{align*}

✓ Mathematica. Time used: 0.189 (sec). Leaf size: 19

ode=(-y[x] + x*D[y[x],x])^2 + x^2*(x - y[x])*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

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