83.9.6 problem 6

Internal problem ID [21964]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. X at page 57
Problem number : 6
Date solved : Thursday, October 02, 2025 at 08:20:14 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 39

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

\begin{align*} y &= -\frac {1}{x} \\ y &= -\frac {2 \operatorname {LambertW}\left (-\frac {\sqrt {x}\, c_1}{2}\right )}{x} \\ y &= -\frac {2 \operatorname {LambertW}\left (\frac {\sqrt {x}\, c_1}{2}\right )}{x} \\ \end{align*}

✓ Mathematica. Time used: 1.922 (sec). Leaf size: 53

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

\begin{align*} y(x)&\to -\frac {1}{x}\\ y(x)&\to -\frac {2 W\left (e^{-1+\frac {9 c_1}{2^{2/3}}} \sqrt {x}\right )}{x}\\ y(x)&\to 0\\ y(x)&\to -\frac {1}{x} \end{align*}

✓ Sympy. Time used: 0.821 (sec). Leaf size: 42

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

\[ \left [ y{\left (x \right )} = - \frac {1}{x}, \ y{\left (x \right )} = - \frac {2 W\left (- \frac {\sqrt {C_{1} x}}{2}\right )}{x}, \ y{\left (x \right )} = - \frac {2 W\left (\frac {\sqrt {C_{1} x}}{2}\right )}{x}\right ] \]