78.6.13 problem 4.d

Internal problem ID [21076]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 7, Nonlinear systems. Problems section 7.11
Problem number : 4.d
Date solved : Thursday, October 02, 2025 at 07:04:37 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 44

ode:=diff(x(t),t) = x(t)-mu*x(t)/(1+x(t)^2); 
dsolve(ode,x(t), singsol=all);
 

\[ x = {\mathrm e}^{\frac {\mu \operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{-2 c_1 \mu +\mu \textit {\_Z} -2 t \mu +2 c_1 +2 t}+\mu -1\right )}{2}-\left (\mu -1\right ) \left (t +c_1 \right )} \]

✓ Mathematica. Time used: 0.307 (sec). Leaf size: 67

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

\begin{align*} x(t)&\to \text {InverseFunction}\left [\frac {2 \log (\text {$\#$1})-\mu \log \left (\text {$\#$1}^2-\mu +1\right )}{2-2 \mu }\&\right ][t+c_1]\\ x(t)&\to 0\\ x(t)&\to -\sqrt {\mu -1}\\ x(t)&\to \sqrt {\mu -1} \end{align*}

✓ Sympy. Time used: 0.693 (sec). Leaf size: 27

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

\[ \frac {\mu \log {\left (- \mu + x^{2}{\left (t \right )} + 1 \right )}}{2 \left (\mu - 1\right )} - t - \frac {\log {\left (x{\left (t \right )} \right )}}{\mu - 1} = C_{1} \]