72.8.6 problem 6

Internal problem ID [19490]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Miscellaneous Problems for Chapter 2. Problems at page 99
Problem number : 6
Date solved : Thursday, October 02, 2025 at 04:32:10 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} x^{2} y^{3}+y&=\left (x^{3} y^{2}-x \right ) y^{\prime } \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 21

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

\[ y = \frac {\sqrt {-\frac {1}{\operatorname {LambertW}\left (-\frac {c_1}{x^{4}}\right )}}}{x} \]

✓ Mathematica. Time used: 5.892 (sec). Leaf size: 60

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

\begin{align*} y(x)&\to -\frac {i}{x \sqrt {W\left (-\frac {e^{-2 c_1}}{x^4}\right )}}\\ y(x)&\to \frac {i}{x \sqrt {W\left (-\frac {e^{-2 c_1}}{x^4}\right )}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.759 (sec). Leaf size: 24

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

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