63.18.6 problem 3(b)

Internal problem ID [13141]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 190
Problem number : 3(b)
Date solved : Monday, March 31, 2025 at 07:35:20 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 )&=x \left (t \right )+y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 39

ode={D[x[t],t]==x[t]-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 e^t (c_1 \cos (t)-c_2 \sin (t)) \\ y(t)\to e^t (c_2 \cos (t)+c_1 \sin (t)) \\ \end{align*}

✓ Sympy. Time used: 0.099 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + 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)
 

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