80.3.5 problem 5

Internal problem ID [21169]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 3. First order nonlinear differential equations. Excercise 3.7 at page 67
Problem number : 5
Date solved : Thursday, October 02, 2025 at 07:15:18 PM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=\frac {x^{2}-x}{2 x-1} \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 16

ode:=diff(x(t),t) = (x(t)^2-x(t))/(2*x(t)-1); 
ic:=[x(0) = 2]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {1}{2}+\frac {\sqrt {1+8 \,{\mathrm e}^{t}}}{2} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 22

ode=D[x[t],t]==(x[t]^2-x[t])/(2*x[t]-1); 
ic={x[0]==2}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{2} \left (\sqrt {8 e^t+1}+1\right ) \end{align*}

✓ Sympy. Time used: 0.485 (sec). Leaf size: 17

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

\[ x{\left (t \right )} = \frac {\sqrt {8 e^{t} + 1}}{2} + \frac {1}{2} \]