65.4.9 problem 9

Internal problem ID [15668]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.2, page 53
Problem number : 9
Date solved : Thursday, October 02, 2025 at 10:22:29 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {x y}{1-y} \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 0.073 (sec). Leaf size: 39

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]-1}{K[1]}dK[1]\&\right ]\left [-\frac {x^2}{2}+c_1\right ]\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.230 (sec). Leaf size: 15

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

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