23.4.169 problem 169

Internal problem ID [6471]
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 : 169
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*} \left (x -y\right ) y^{\prime \prime }&=\left (1+y^{\prime }\right ) \left (1+{y^{\prime }}^{2}\right ) \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 107

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

\begin{align*} y &= x +\operatorname {RootOf}\left (-x -\int _{}^{\textit {\_Z}}\frac {c_1^{2} \textit {\_f}^{2}-1}{c_1^{2} \textit {\_f}^{2}+\sqrt {-c_1^{2} \textit {\_f}^{2}+2}\, c_1 \textit {\_f} -2}d \textit {\_f} +c_2 \right ) \\ y &= x +\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}-\frac {c_1^{2} \textit {\_f}^{2}-1}{-2+c_1^{2} \textit {\_f}^{2}-\sqrt {-c_1^{2} \textit {\_f}^{2}+2}\, c_1 \textit {\_f}}d \textit {\_f} +c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 33.419 (sec). Leaf size: 18884

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

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - y(x))*Derivative(y(x), (x, 2)) - (Derivative(y(x), x) + 1)*(Derivative(y(x), x)**2 + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE (-27*x*Derivative(y(x), (x, 2))/2 + sqrt((-27*x*Derivative(y(x),