23.4.224 problem 224

Internal problem ID [6526]
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 : 224
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*} x^{2} \left (x -y\right ) y^{\prime \prime }&=\left (-y+x y^{\prime }\right )^{2} \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 49

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

\begin{align*} y &= x \\ y &= x -\sqrt {\left (-2 c_1 +1\right ) x^{2}-2 c_2 x} \\ y &= x +\sqrt {\left (-2 c_1 +1\right ) x^{2}-2 c_2 x} \\ \end{align*}

✓ Mathematica. Time used: 0.342 (sec). Leaf size: 56

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

\begin{align*} y(x)&\to x-\sqrt {x} \sqrt {x+2 c_2 x-2 c_1}\\ y(x)&\to x+\sqrt {x} \sqrt {x+2 c_2 x-2 c_1} \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)