65.15.5 problem 28.6 (iii)

Internal problem ID [13758]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 28, Distinct real eigenvalues. Exercises page 282
Problem number : 28.6 (iii)
Date solved : Wednesday, March 05, 2025 at 10:15:01 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 24

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

\begin{align*} x \left (t \right ) &= c_{1} +{\mathrm e}^{-3 t} c_{2} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-3 t} c_{2}}{2}+c_{1} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 71

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

\begin{align*} x(t)\to \frac {1}{3} e^{-3 t} \left (c_1 \left (e^{3 t}+2\right )+2 c_2 \left (e^{3 t}-1\right )\right ) \\ y(t)\to \frac {1}{3} e^{-3 t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (2 e^{3 t}+1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.076 (sec). Leaf size: 22

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

\[ \left [ x{\left (t \right )} = C_{1} - 2 C_{2} e^{- 3 t}, \ y{\left (t \right )} = C_{1} + C_{2} e^{- 3 t}\right ] \]