53.4.5 problem 5

Internal problem ID [8493]
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 : 5
Date solved : Sunday, March 30, 2025 at 01:12:00 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} y^{2} y^{\prime \prime }+{y^{\prime }}^{3}&=0 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 29

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

\begin{align*} y &= 0 \\ y &= c_1 \\ y &= -\frac {\operatorname {LambertW}\left (-c_1 \,{\mathrm e}^{-c_2 -x}\right )}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.849 (sec). Leaf size: 37

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

\[ y(x)\to c_2 \left (1+\frac {1}{\text {InverseFunction}\left [-\frac {1}{\text {$\#$1}}-\log (\text {$\#$1})+\log (\text {$\#$1}+1)\&\right ][-x+c_1]}\right ) \]

✗ Sympy

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

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