41.2.54 problem 50

Internal problem ID [8756]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 50
Date solved : Tuesday, September 30, 2025 at 05:50:17 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 0.108 (sec). Leaf size: 63

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

\[ \text {Solve}\left [\int _1^{-\frac {2 x y(x)+1}{\sqrt [3]{2} (x y(x)-1)}}\frac {1}{K[1]^3-\frac {3 K[1]}{2^{2/3}}+1}dK[1]+\frac {2}{9} 2^{2/3} \log (x)=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.526 (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)
 

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