60.7.112 problem 1725 (book 6.134)

Internal problem ID [11662]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1725 (book 6.134)
Date solved : Sunday, March 30, 2025 at 08:37:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 21

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

\[ y = \frac {c_2^{2}-c_2 x +c_1}{c_2 -x} \]

✓ Mathematica. Time used: 0.425 (sec). Leaf size: 123

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

\[ \text {Solve}\left [\left \{x=\frac {1}{2} \int \frac {(1-K[4]) \exp \left (-\int _1^{K[4]}\frac {1-K[3]}{2 K[3] (K[3]+1)}dK[3]-c_1\right )}{(K[4]-1) K[4] (K[4]+1)} \, dK[4]+c_2,y(x)=x-\exp \left (-\int _1^{K[4]}\frac {1-K[3]}{2 K[3] (K[3]+1)}dK[3]-c_1\right )\right \},\{y(x),K[4]\}\right ] \]

✗ Sympy

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

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