23.1.577 problem 570

Internal problem ID [5184]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 570
Date solved : Tuesday, September 30, 2025 at 11:51:37 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 18

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

\[ y = -\frac {1}{\operatorname {LambertW}\left (-\frac {c_1}{x^{2}}\right ) x} \]

✓ Mathematica. Time used: 5.071 (sec). Leaf size: 35

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

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

✓ Sympy. Time used: 0.515 (sec). Leaf size: 19

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

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