60.7.152 problem 1769 (book 6.178)

Internal problem ID [11702]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1769 (book 6.178)
Date solved : Wednesday, March 05, 2025 at 02:41:12 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 49

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

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

✓ Mathematica. Time used: 10.945 (sec). Leaf size: 99

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

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x + sqrt((x + y(x))*(-4*x**2*Derivative(y(x), (x, 2)) + x + y(x))) + y(x))/(2*x) cannot be solved by the factorable group method