64.25.8 problem 8

Internal problem ID [13632]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 13, Limit cycles and periodic solutions. Section 13.4, Exercises page 706
Problem number : 8
Date solved : Monday, March 31, 2025 at 08:03:23 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-x \left (t \right )^{2}\\ \frac {d}{d t}y \left (t \right )&=2 y \left (t \right )-y \left (t \right )^{2} \end{align*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 32

ode:=[diff(x(t),t) = x(t)-x(t)^2, diff(y(t),t) = 2*y(t)-y(t)^2]; 
dsolve(ode);
 

\begin{align*} \left \{x \left (t \right ) &= \frac {1}{1+{\mathrm e}^{-t} c_2}\right \} \\ \left \{y \left (t \right ) &= \frac {2}{1+2 \,{\mathrm e}^{-2 t} c_1}\right \} \\ \end{align*}

✓ Mathematica. Time used: 0.448 (sec). Leaf size: 238

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

\begin{align*} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1]}dK[1]\&\right ][-t+c_1] \\ y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[2]-2) K[2]}dK[2]\&\right ][-t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1]}dK[1]\&\right ][-t+c_1] \\ y(t)\to 0 \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1]}dK[1]\&\right ][-t+c_1] \\ y(t)\to 2 \\ x(t)\to 0 \\ y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[2]-2) K[2]}dK[2]\&\right ][-t+c_2] \\ x(t)\to 0 \\ y(t)\to 0 \\ x(t)\to 0 \\ y(t)\to 2 \\ x(t)\to 1 \\ y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[2]-2) K[2]}dK[2]\&\right ][-t+c_2] \\ x(t)\to 1 \\ y(t)\to 0 \\ x(t)\to 1 \\ y(t)\to 2 \\ \end{align*}

✗ Sympy

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

NotImplementedError :