89.33.39 problem 42

Internal problem ID [25023]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 17. Special Equations of order Two. Exercises at page 251
Problem number : 42
Date solved : Thursday, October 02, 2025 at 11:47:24 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} 3 y y^{\prime } y^{\prime \prime }&=-1+{y^{\prime }}^{3} \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 112

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

\begin{align*} \frac {3 \left (c_1 y+1\right )^{{2}/{3}}+\left (-2 x -2 c_2 \right ) c_1}{2 c_1} &= 0 \\ \frac {3 i \left (c_1 y+1\right )^{{2}/{3}}-\left (x +c_2 \right ) c_1 \sqrt {3}+i \left (x +c_2 \right ) c_1}{c_1 \left (-i+\sqrt {3}\right )} &= 0 \\ \frac {-3 i \left (c_1 y+1\right )^{{2}/{3}}-\left (x +c_2 \right ) c_1 \sqrt {3}-i \left (x +c_2 \right ) c_1}{c_1 \left (\sqrt {3}+i\right )} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 36.953 (sec). Leaf size: 126

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

\begin{align*} y(x)&\to \frac {1}{9} e^{-3 c_1} \left (-9+2 \sqrt {6} \left (e^{3 c_1} (x+c_2)\right ){}^{3/2}\right )\\ y(x)&\to \frac {1}{9} e^{-3 c_1} \left (-9+2 \sqrt {6} \left (-\sqrt [3]{-1} e^{3 c_1} (x+c_2)\right ){}^{3/2}\right )\\ y(x)&\to \frac {1}{9} e^{-3 c_1} \left (-9+2 \sqrt {6} \left ((-1)^{2/3} e^{3 c_1} (x+c_2)\right ){}^{3/2}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x)*Derivative(y(x), x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**3 + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(sqrt(-64*y(x)**3*Derivative(y(x), (x, 2))**3/27 + 1/4) + 1/2)*