23.4.223 problem 223

Internal problem ID [6525]
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 : 223
Date solved : Friday, October 03, 2025 at 02:09:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 2 \left (1-y\right )^{2} y-2 x \left (1-y\right ) y^{\prime }+2 x^{2} {y^{\prime }}^{2}+x^{2} \left (1-y\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 30

ode:=2*(1-y(x))^2*y(x)-2*x*(1-y(x))*diff(y(x),x)+2*x^2*diff(y(x),x)^2+x^2*(1-y(x))*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.465 (sec). Leaf size: 27

ode=2*(1 - y[x])^2*y[x] - 2*x*(1 - y[x])*D[y[x],x] + 2*x^2*D[y[x],x]^2 + x^2*(1 - y[x])*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 1+\frac {1}{c_2 x^2-c_1 x-1}\\ y(x)&\to 1 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - y(x))*Derivative(y(x), (x, 2)) + 2*x**2*Derivative(y(x), x)**2 - 2*x*(1 - y(x))*Derivative(y(x), x) + 2*(1 - y(x))**2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt((y(x) - 1)*(2*x**2*Derivative(y(x),