26.10.6 problem Exercise 35.6, page 504

Internal problem ID [7135]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 8. Special second order equations. Lesson 35. Independent variable x absent
Problem number : Exercise 35.6, page 504
Date solved : Tuesday, September 30, 2025 at 04:21:58 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 59

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

\begin{align*} y &= -1 \\ y &= -\frac {\sqrt {-2 c_1 x -2 c_2}-1}{\sqrt {-2 c_1 x -2 c_2}} \\ y &= -\frac {\sqrt {-2 c_1 x -2 c_2}+1}{\sqrt {-2 c_1 x -2 c_2}} \\ \end{align*}

✓ Mathematica. Time used: 0.94 (sec). Leaf size: 107

ode=(y[x]+1)*D[y[x],{x,2}]==3*(D[y[x],x])^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {2 c_1 x+\sqrt {2} \sqrt {-c_1 (x+c_2)}+2 c_2 c_1}{2 c_1 (x+c_2)}\\ y(x)&\to \frac {-2 c_1 x+\sqrt {2} \sqrt {-c_1 (x+c_2)}-2 c_2 c_1}{2 c_1 (x+c_2)}\\ y(x)&\to -1\\ y(x)&\to \text {Indeterminate} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(3)*sqrt((y(x) + 1)*Derivative(y(x), (x, 2)))/3 + Derivative(y(x), x) cannot be solved by the factorable group method