23.4.216 problem 216

Internal problem ID [6518]
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 : 216
Date solved : Tuesday, September 30, 2025 at 03:02:36 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*} \left (x -y\right ) y^{\prime }+x {y^{\prime }}^{2}+x \left (x +y\right ) y^{\prime \prime }&=y \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 49

ode:=(x-y(x))*diff(y(x),x)+x*diff(y(x),x)^2+x*(x+y(x))*diff(diff(y(x),x),x) = y(x); 
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: 0.514 (sec). Leaf size: 53

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

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