26.10.16 problem Exercise 35.16, page 504

Internal problem ID [7145]
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 35.16, page 504
Date solved : Tuesday, September 30, 2025 at 04:23:18 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*}

With initial conditions

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

✓ Maple. Time used: 0.446 (sec). Leaf size: 15

ode:=(y(x)+1)*diff(diff(y(x),x),x) = 3*diff(y(x),x)^2; 
ic:=[y(1) = 0, D(y)(1) = -1/2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {-x +\sqrt {x}}{x} \]

✓ Mathematica. Time used: 0.764 (sec). Leaf size: 572

ode=(y[x]+1)*D[y[x],{x,2}]==3*(D[y[x],x])^2; 
ic={y[1]==0,Derivative[1][y][0] ==-1/2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {6 \left (\left (-12+3\ 2^{2/3} \sqrt [3]{27-3 \sqrt {69}}-\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+3\ 2^{2/3} \sqrt [3]{3 \left (9+\sqrt {69}\right )}-\sqrt [3]{2} \left (3 \left (9+\sqrt {69}\right )\right )^{2/3}\right ) x+3 \sqrt {2} \sqrt {\left (12-3\ 2^{2/3} \sqrt [3]{27-3 \sqrt {69}}+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}-3\ 2^{2/3} \sqrt [3]{3 \left (9+\sqrt {69}\right )}+\sqrt [3]{2} \left (3 \left (9+\sqrt {69}\right )\right )^{2/3}\right ) x-\sqrt [3]{2} \left (3 \left (9+\sqrt {69}\right )\right )^{2/3}+2\ 2^{2/3} \sqrt [3]{3 \left (9+\sqrt {69}\right )}+\sqrt [3]{9-\sqrt {69}} \left (9+\sqrt {69}\right )^{2/3}+\left (9-\sqrt {69}\right )^{2/3} \sqrt [3]{9+\sqrt {69}}-\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+2\ 2^{2/3} \sqrt [3]{27-3 \sqrt {69}}+6}+\sqrt [3]{2} \left (3 \left (9+\sqrt {69}\right )\right )^{2/3}-2\ 2^{2/3} \sqrt [3]{3 \left (9+\sqrt {69}\right )}-\sqrt [3]{9-\sqrt {69}} \left (9+\sqrt {69}\right )^{2/3}-\left (9-\sqrt {69}\right )^{2/3} \sqrt [3]{9+\sqrt {69}}+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}-2\ 2^{2/3} \sqrt [3]{27-3 \sqrt {69}}-6\right )}{\left (12-3\ 2^{2/3} \sqrt [3]{27-3 \sqrt {69}}+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}-3\ 2^{2/3} \sqrt [3]{3 \left (9+\sqrt {69}\right )}+\sqrt [3]{2} \left (3 \left (9+\sqrt {69}\right )\right )^{2/3}\right ) \left (6 x+2^{2/3} \sqrt [3]{3} \left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right )\right )} \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 = {y(1): 0, Subs(Derivative(y(x), x), x, 1): -1/2} 
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