54.7.111 problem 1724 (book 6.133)

Internal problem ID [12960]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1724 (book 6.133)
Date solved : Wednesday, October 01, 2025 at 02:55:19 AM
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*} y^{\prime \prime } \left (x +y\right )+{y^{\prime }}^{2}-y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.090 (sec). Leaf size: 18

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

\[ y = \sqrt {-2 x +c_1}\, c_2 -c_1 +x \]

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 99

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

\[ \text {Solve}\left [\left \{x=c_2-\int \frac {\exp \left (-\int _1^{K[2]}\frac {K[1]+1}{(K[1]-1) K[1]}dK[1]-c_1\right )}{(K[2]-1) K[2]} \, dK[2],y(x)=-x+\exp \left (-\int _1^{K[2]}\frac {K[1]+1}{(K[1]-1) K[1]}dK[1]-c_1\right )\right \},\{y(x),K[2]\}\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(-4*x*Derivative(y(x), (x, 2)) - 4*y(x)*Derivative(y(x), (x, 2)) + 1)/2 + Derivative(y(x), x) - 1/2 cannot be solved by the factorable group method