63.20.5 problem 4

Internal problem ID [13148]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 218
Problem number : 4
Date solved : Wednesday, March 05, 2025 at 09:18:17 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )-4 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 3\\ y \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.068 (sec). Leaf size: 33

ode:=[diff(x(t),t) = x(t)-2*y(t), diff(y(t),t) = 3*x(t)-4*y(t)]; 
ic:=x(0) = 3y(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= 7 \,{\mathrm e}^{-t}-4 \,{\mathrm e}^{-2 t} \\ y &= 7 \,{\mathrm e}^{-t}-6 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 34

ode={D[x[t],t]==x[t]-2*y[t],D[y[t],t]==3*x[t]-4*y[t]}; 
ic={x[0]==3,y[0]==1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-2 t} \left (7 e^t-4\right ) \\ y(t)\to e^{-2 t} \left (7 e^t-6\right ) \\ \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-3*x(t) + 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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