63.22.9 problem 5

Internal problem ID [13160]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 237
Problem number : 5
Date solved : Wednesday, March 05, 2025 at 09:18:29 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = -1 \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 13

ode:=[diff(x(t),t) = 2*x(t)+y(t), diff(y(t),t) = -x(t)]; 
ic:=x(0) = 1y(0) = -1; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 16

ode={D[x[t],t]==2*x[t]+y[t],D[y[t],t]==-x[t]+0*y[t]}; 
ic={x[0]==1,y[0]==-1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.084 (sec). Leaf size: 31

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

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