9.5.6 problem 6

Internal problem ID [2942]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 9, page 38
Problem number : 6
Date solved : Tuesday, September 30, 2025 at 06:11:16 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} \left (x^{3} y^{3}-1\right ) y^{\prime }+x^{2} y^{4}&=0 \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 32

ode:=(x^3*y(x)^3-1)*diff(y(x),x)+x^2*y(x)^4 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-c_1}}{\left (-\frac {x^{3} {\mathrm e}^{-3 c_1}}{\operatorname {LambertW}\left (-x^{3} {\mathrm e}^{-3 c_1}\right )}\right )^{{1}/{3}}} \]

✓ Mathematica. Time used: 2.863 (sec). Leaf size: 90

ode=(x^3*y[x]^3-1)*D[y[x],x]+x^2*y[x]^4==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt [3]{W\left (-e^{-3 c_1} x^3\right )}}{x}\\ y(x)&\to \frac {\sqrt [3]{-1} \sqrt [3]{W\left (-e^{-3 c_1} x^3\right )}}{x}\\ y(x)&\to -\frac {(-1)^{2/3} \sqrt [3]{W\left (-e^{-3 c_1} x^3\right )}}{x}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.326 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*y(x)**4 + (x**3*y(x)**3 - 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - \frac {x^{3} y^{3}{\left (x \right )}}{3} - \log {\left (x \right )} + \log {\left (x y{\left (x \right )} \right )} = C_{1} \]