57.21.1 problem 1(a)

Internal problem ID [14510]
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(a)
Date solved : Thursday, October 02, 2025 at 09:38:01 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=-3 x+y \left (t \right )\\ y^{\prime }\left (t \right )&=-3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.146 (sec). Leaf size: 23

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

\begin{align*} x \left (t \right ) &= \left (c_2 t +c_1 \right ) {\mathrm e}^{-3 t} \\ y \left (t \right ) &= c_2 \,{\mathrm e}^{-3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 29

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

\begin{align*} x(t)&\to e^{-3 t} (c_2 t+c_1)\\ y(t)&\to c_2 e^{-3 t} \end{align*}

✓ Sympy. Time used: 0.042 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) - y(t) + Derivative(x(t), t),0),Eq(3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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