89.6.47 problem 48

Internal problem ID [24430]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Miscellaneous Exercises at page 45
Problem number : 48
Date solved : Thursday, October 02, 2025 at 10:29:22 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y^{3}-x^{3}&=x y \left (y y^{\prime }+x \right ) \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 30

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

\[ y = x \left ({\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{2 \textit {\_Z}}-4 \,{\mathrm e}^{\textit {\_Z}}+2 c_1 +2 \textit {\_Z} +2 \ln \left (x \right )+3\right )}-1\right ) \]

✓ Mathematica. Time used: 0.088 (sec). Leaf size: 45

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

\[ \text {Solve}\left [\frac {1}{2} \left (\frac {y(x)}{x}+1\right )^2-2 \left (\frac {y(x)}{x}+1\right )+\log \left (\frac {y(x)}{x}+1\right )=-\log (x)+c_1,y(x)\right ] \]

✗ Sympy

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

TypeError : cannot determine truth value of Relational: -1 < 2*x**2