23.4.168 problem 168

Internal problem ID [6470]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 168
Date solved : Friday, October 03, 2025 at 02:09:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.512 (sec). Leaf size: 40

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]
 

\begin{align*} y(x)&\to -c_2\\ y(x)&\to -\frac {e^{-c_1}}{x+c_2}-c_2\\ y(x)&\to -c_2 \end{align*}

✗ 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) + 1)*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