54.7.155 problem 1772 (book 6.181)

Internal problem ID [13004]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1772 (book 6.181)
Date solved : Friday, October 03, 2025 at 03:58:16 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 (x y^{\prime }-y\right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 29

ode:=x^2*(x+y(x))*diff(diff(y(x),x),x)-(-y(x)+x*diff(y(x),x))^2 = 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: 0.542 (sec). Leaf size: 42

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 \left (-1+e^{c_1-\frac {e^{c_2}}{x}}\right )\\ y(x)&\to -x\\ y(x)&\to \left (-1+e^{c_1}\right ) 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))) + y(x))/x cannot be solved by the factorable group method