63.19.5 problem 1(e)

Internal problem ID [13140]
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(e)
Date solved : Wednesday, March 05, 2025 at 09:18:07 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 )&=-2 x \left (t \right )+4 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 26

ode:=[diff(x(t),t) = x(t)-2*y(t), diff(y(t),t) = -2*x(t)+4*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{5 t} \\ y &= -2 c_{2} {\mathrm e}^{5 t}+\frac {c_{1}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 62

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

\begin{align*} x(t)\to \frac {1}{5} \left (c_1 \left (e^{5 t}+4\right )-2 c_2 \left (e^{5 t}-1\right )\right ) \\ y(t)\to \frac {1}{5} \left (c_2 \left (4 e^{5 t}+1\right )-2 c_1 \left (e^{5 t}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.082 (sec). Leaf size: 24

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(2*x(t) - 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 2 C_{1} - \frac {C_{2} e^{5 t}}{2}, \ y{\left (t \right )} = C_{1} + C_{2} e^{5 t}\right ] \]