38.5.44 problem 40 (c)

Internal problem ID [8392]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 40 (c)
Date solved : Tuesday, September 30, 2025 at 05:34:34 PM
CAS classification : [_separable]

\begin{align*} x y^{\prime }&=y^{2}-y \end{align*}

With initial conditions

\begin{align*} y \left (\frac {1}{2}\right )&={\frac {1}{2}} \\ \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 11

ode:=x*diff(y(x),x) = y(x)^2-y(x); 
ic:=[y(1/2) = 1/2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {1}{2 x +1} \]

✓ Mathematica. Time used: 0.143 (sec). Leaf size: 12

ode=x*D[y[x],x]==y[x]^2-y[x]; 
ic={y[1/2]==1/2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2 x+1} \end{align*}

✓ Sympy. Time used: 0.164 (sec). Leaf size: 12

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

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