53.4.39 problem 42

Internal problem ID [8527]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 42
Date solved : Wednesday, March 05, 2025 at 06:02:38 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.060 (sec). Leaf size: 114

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

✓ Mathematica. Time used: 44.385 (sec). Leaf size: 126

ode=3*y[x]*D[y[x],x]*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(3*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(-y(x)**3*Derivative(y(x), (x, 2))**3 + 1/4) + 1/2)**(1/3) + Derivative(y(x), x) - y(x)*Derivative(y(x), (x, 2))/(sqrt(-y(x)**3*Derivative(y(x), (x, 2))**3 + 1/4) + 1/2)**(1/3) cannot be solved by the factorable group method