58.1.50 problem 50

Internal problem ID [9121]
Book : Second order enumerated odes
Section : section 1
Problem number : 50
Date solved : Sunday, March 30, 2025 at 02:08:05 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 27

ode:=y(x)*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 {c_2 +x}{\operatorname {LambertW}\left (\left (c_2 +x \right ) {\mathrm e}^{c_1 -1}\right )} \\ \end{align*}

✓ Mathematica. Time used: 60.104 (sec). Leaf size: 26

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

\[ y(x)\to \frac {x+c_2}{W\left (e^{-1-c_1} (x+c_2)\right )} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*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)*Derivative(y(x), (x, 2)))**(1/3)/2 - sqrt(3)*I*(-y(x)*Derivative(y(x), (x, 2)))**(1/3)/2 + Derivative(y(x), x) cannot be solved by the factorable group method