23.4.239 problem 239

Internal problem ID [6541]
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 : 239
Date solved : Tuesday, September 30, 2025 at 03:03:59 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 21

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

\begin{align*} y &= -i \\ y &= i \\ y &= \tan \left (\ln \left (c_1 x +c_2 \right )\right ) \\ \end{align*}

✓ Mathematica. Time used: 3.036 (sec). Leaf size: 97

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

\begin{align*} y(x)&\to -\frac {i \left (-1+c_1{}^{2 i} (x+c_2){}^{2 i}\right )}{1+c_1{}^{2 i} (x+c_2){}^{2 i}}\\ y(x)&\to \frac {i \left (e^{2 \arg (x+c_2)}-e^{2 i \text {Interval}[\{0,\pi \}]}\right )}{e^{2 i \text {Interval}[\{0,\pi \}]}+e^{2 \arg (x+c_2)}} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt((y(x)**2 + 1)*Derivative(y(x), (x, 2))/(2*y(x) - 1)) + Der