63.18.3 problem 2(c)

Internal problem ID [13138]
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 : 2(c)
Date solved : Monday, March 31, 2025 at 07:35:16 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-3 x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.105 (sec). Leaf size: 19

ode:=[diff(x(t),t) = -3*x(t), diff(y(t),t) = 2*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_2 \,{\mathrm e}^{-3 t} \\ y \left (t \right ) &= c_1 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 65

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

\begin{align*} x(t)\to c_1 e^{-3 t} \\ y(t)\to c_2 e^{3 t} \\ x(t)\to c_1 e^{-3 t} \\ y(t)\to 0 \\ x(t)\to 0 \\ y(t)\to c_2 e^{3 t} \\ x(t)\to 0 \\ y(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.068 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) + Derivative(x(t), t),0),Eq(-2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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