14.17.15 problem 15

Internal problem ID [3869]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.5 (Defective coefficient matrix), page 619
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 06:58:18 AM
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*}

With initial conditions

\begin{align*} x_{1} \left (0\right )&=0 \\ x_{2} \left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.114 (sec). Leaf size: 21

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)]; 
ic:=[x__1(0) = 0, x__2(0) = -1]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 24

ode={D[x1[t],t]==-2*x1[t]-1*x2[t],D[x2[t],t]==1*x1[t]-4*x2[t]}; 
ic={x1[0]==0,x2[0]==-1}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.054 (sec). Leaf size: 36

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)
 

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