25.3.9 problem 7.1

Internal problem ID [6874]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 4
Problem number : 7.1
Date solved : Tuesday, September 30, 2025 at 03:59:50 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 42

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

\begin{align*} y &= -\frac {1}{x} \\ y &= \frac {{\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}}-2 \ln \left (x \right ) {\mathrm e}^{\textit {\_Z}}+2 c_1 \,{\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+1\right )}}{x} \\ \end{align*}

✓ Mathematica. Time used: 0.15 (sec). Leaf size: 35

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

\begin{align*} y(x)&\to -\frac {1}{x}\\ \text {Solve}\left [x y(x)-\frac {1}{x y(x)}-2 \log (y(x))=c_1,y(x)\right ] \end{align*}

✓ Sympy. Time used: 1.046 (sec). Leaf size: 32

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

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