85.14.8 problem 6

Internal problem ID [22524]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. C Exercises at page 41
Problem number : 6
Date solved : Thursday, October 02, 2025 at 08:46:22 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.069 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 6.196 (sec). Leaf size: 33

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

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

✓ Sympy. Time used: 0.330 (sec). Leaf size: 12

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

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