63.19.3 problem 1(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 202
Problem number : 1(c)
Date solved : Wednesday, March 05, 2025 at 09:18:05 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 22

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 35

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

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

✓ Sympy. Time used: 0.072 (sec). Leaf size: 22

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

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