63.21.2 problem 1(b)

Internal problem ID [13151]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 225
Problem number : 1(b)
Date solved : Wednesday, March 05, 2025 at 09:18:20 PM
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 )+3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 29

ode:=[diff(x(t),t) = x(t)-y(t), diff(y(t),t) = x(t)+3*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 42

ode={D[x[t],t]==x[t]-y[t],D[y[t],t]==x[t]+3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.111 (sec). Leaf size: 37

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) - 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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