64.1.8 problem 2.2 (ii)

Internal problem ID [15585]
Book : Nonlinear Ordinary Differential Equations by D.W.Jordna and P.Smith. 4th edition 1999. Oxford Univ. Press. NY
Section : Chapter 2. Plane autonomous systems and linearization. Problems page 79
Problem number : 2.2 (ii)
Date solved : Thursday, October 02, 2025 at 10:20:27 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.110 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 59

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

\begin{align*} x(t)&\to e^{-t} \left (c_1 \left (2 e^t-1\right )-c_2 \left (e^t-1\right )\right )\\ y(t)&\to e^{-t} \left (2 c_1 \left (e^t-1\right )-c_2 \left (e^t-2\right )\right ) \end{align*}

✓ Sympy. Time used: 0.040 (sec). Leaf size: 19

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

\[ \left [ x{\left (t \right )} = C_{1} + \frac {C_{2} e^{- t}}{2}, \ y{\left (t \right )} = C_{1} + C_{2} e^{- t}\right ] \]