23.4.167 problem 167

Internal problem ID [6469]
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 : 167
Date solved : Tuesday, September 30, 2025 at 03:01:36 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*} -y^{\prime }+{y^{\prime }}^{2}+\left (x +y\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 13.215 (sec). Leaf size: 122

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]
 

\begin{align*} y(x)&\to x-\frac {e^{-2 c_1} \sqrt {e^{2 c_1} \left (1+4 e^{c_1} (x+c_2)\right )}}{\sqrt {2}}+\frac {e^{-c_1}}{2}+2 c_2\\ y(x)&\to x+\frac {e^{-2 c_1} \sqrt {e^{2 c_1} \left (1+4 e^{c_1} (x+c_2)\right )}}{\sqrt {2}}+\frac {e^{-c_1}}{2}+2 c_2\\ y(x)&\to x+2 c_2 \end{align*}

✗ 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