23.4.261 problem 261

Internal problem ID [6563]
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 : 261
Date solved : Tuesday, September 30, 2025 at 03:04:30 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} \left (1-3 y^{2}\right ) {y^{\prime }}^{2}+y \left (1+y^{2}\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 80

ode:=(1-3*y(x)^2)*diff(y(x),x)^2+y(x)*(1+y(x)^2)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -i \\ y &= i \\ y &= 0 \\ y &= -\frac {\sqrt {2}\, \sqrt {-2 \left (c_1 x +c_2 +\frac {1}{2}\right ) \left (c_1 x +c_2 \right )}}{2 c_1 x +2 c_2} \\ y &= \frac {\sqrt {2}\, \sqrt {-2 \left (c_1 x +c_2 +\frac {1}{2}\right ) \left (c_1 x +c_2 \right )}}{2 c_1 x +2 c_2} \\ \end{align*}

✓ Mathematica. Time used: 1.202 (sec). Leaf size: 223

ode=(1 - 3*y[x]^2)*D[y[x],x]^2 + y[x]*(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 {\sqrt {-2 c_1 x-1-2 c_2 c_1}}{\sqrt {2} \sqrt {c_1 (x+c_2)}}\\ y(x)&\to \frac {\sqrt {-2 c_1 x-1-2 c_2 c_1}}{\sqrt {2} \sqrt {c_1 (x+c_2)}}\\ y(x)&\to \text {Indeterminate}\\ y(x)&\to \frac {\sqrt {-c_1}}{\sqrt {c_1}}\\ y(x)&\to \frac {\sqrt {c_1}}{\sqrt {-c_1}}\\ y(x)&\to -\frac {\sqrt {-x-c_2}}{\sqrt {x+c_2}}\\ y(x)&\to \frac {\sqrt {-x-c_2}}{\sqrt {x+c_2}}\\ y(x)&\to -\frac {\sqrt {x+c_2}}{\sqrt {-x-c_2}}\\ y(x)&\to \frac {\sqrt {x+c_2}}{\sqrt {-x-c_2}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - 3*y(x)**2)*Derivative(y(x), x)**2 + (y(x)**2 + 1)*y(x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt((y(x)**2 + 1)*y(x)*Derivative(y(x), (x, 2))/(3*y(x)**2 - 1