54.7.154 problem 1771 (book 6.180)

Internal problem ID [13003]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1771 (book 6.180)
Date solved : Friday, October 03, 2025 at 03:58:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 30

ode:=x^2*(y(x)-1)*diff(diff(y(x),x),x)-2*x^2*diff(y(x),x)^2-2*x*(y(x)-1)*diff(y(x),x)-2*y(x)*(y(x)-1)^2 = 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

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]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(y(x) - 1)*Derivative(y(x), (x, 2)) - 2*x**2*Derivative(y(x), x)**2 - 2*x*(y(x) - 1)*Derivative(y(x), x) - 2*(y(x) - 1)**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), (x, 2)) - 4*y(x)**2 + 5*y(x) - 1)) - y(x) + 1)/(2*x) cannot be solved by the factorable group method