65.3.12 problem 8.8

Internal problem ID [13659]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 8, Separable equations. Exercises page 72
Problem number : 8.8
Date solved : Monday, March 31, 2025 at 08:04:25 AM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=-x \left (k^{2}+x^{2}\right ) \end{align*}

With initial conditions

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

✗ Maple

ode:=diff(x(t),t) = -x(t)*(k^2+x(t)^2); 
ic:=x(0) = x__0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.234 (sec). Leaf size: 54

ode=D[x[t],t]==-x[t]*(k^2+x[t]^2); 
ic={x[0]==x0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \text {InverseFunction}\left [\int _0^{\text {$\#$1}}\frac {1}{K[1] \left (k^2+K[1]^2\right )}dK[1]\&\right ]\left [\int _0^{\text {x0}}\frac {1}{K[1]^3+k^2 K[1]}dK[1]-t\right ] \]

✗ Sympy

from sympy import * 
t = symbols("t") 
k = symbols("k") 
x = Function("x") 
ode = Eq((k**2 + x(t)**2)*x(t) + Derivative(x(t), t),0) 
ics = {x(0): x__0} 
dsolve(ode,func=x(t),ics=ics)
 

NotImplementedError : Initial conditions produced too many solutions for constants