81.5.3 problem 6-3

Internal problem ID [21541]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 6. Method of grouping. Page 96.
Problem number : 6-3
Date solved : Thursday, October 02, 2025 at 07:47:17 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} \left (x^{3}-y\right ) y-x \left (x^{3}+y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.211 (sec). Leaf size: 41

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

\begin{align*} y &= \frac {\left (c_1 -\sqrt {x^{4}+c_1^{2}}\right ) c_1}{x} \\ y &= \frac {\left (c_1 +\sqrt {x^{4}+c_1^{2}}\right ) c_1}{x} \\ \end{align*}

✓ Mathematica. Time used: 0.592 (sec). Leaf size: 73

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

\begin{align*} y(x)&\to \frac {x^4}{-x+\frac {\sqrt {1+c_1 x^4}}{\sqrt {\frac {1}{x^2}}}}\\ y(x)&\to -\frac {x^4}{x+\frac {\sqrt {1+c_1 x^4}}{\sqrt {\frac {1}{x^2}}}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 5.128 (sec). Leaf size: 48

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

\[ \left [ y{\left (x \right )} = \frac {\left (1 - \sqrt {2 x^{4} e^{C_{1}} + 1}\right ) e^{- C_{1}}}{2 x}, \ y{\left (x \right )} = \frac {\left (\sqrt {2 x^{4} e^{C_{1}} + 1} + 1\right ) e^{- C_{1}}}{2 x}\right ] \]