9.6.1 problem problem 1

Internal problem ID [1008]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 1
Date solved : Saturday, March 29, 2025 at 10:36:46 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-2 x_{1} \left (t \right )+x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )-4 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.120 (sec). Leaf size: 31

ode:=[diff(x__1(t),t) = -2*x__1(t)+x__2(t), diff(x__2(t),t) = -x__1(t)-4*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_2 t +c_1 \right ) \\ x_{2} \left (t \right ) &= -{\mathrm e}^{-3 t} \left (c_2 t +c_1 -c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 42

ode={D[ x1[t],t]==-2*x1[t]+1*x2[t],D[ x2[t],t]==-1*x1[t]-4*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-3 t} (c_1 (t+1)+c_2 t) \\ \text {x2}(t)\to e^{-3 t} (c_2-(c_1+c_2) t) \\ \end{align*}

✓ Sympy. Time used: 0.092 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(2*x__1(t) - x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t) + 4*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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