14.17.16 problem 16

Internal problem ID [3870]
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 : 16
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 )+4 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{1} \left (t \right )-3 x_{2} \left (t \right )+2 x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right )&=-2 \\ x_{2} \left (0\right )&=1 \\ x_{3} \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.183 (sec). Leaf size: 36

ode:=[diff(x__1(t),t) = -2*x__1(t)-x__2(t)+4*x__3(t), diff(x__2(t),t) = -x__2(t), diff(x__3(t),t) = -x__1(t)-3*x__2(t)+2*x__3(t)]; 
ic:=[x__1(0) = -2, x__2(0) = 1, x__3(0) = 1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x_{1} \left (t \right ) &= -9 \,{\mathrm e}^{-t}-2 t +7 \\ x_{2} \left (t \right ) &= {\mathrm e}^{-t} \\ x_{3} \left (t \right ) &= -2 \,{\mathrm e}^{-t}+3-t \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 40

ode={D[x1[t],t]==-2*x1[t]-x2[t]+4*x3[t],D[x2[t],t]==-x2[t],D[x3[t],t]==-x1[t]-3*x2[t]+2*x3[t]}; 
ic={x1[0]==-2,x2[0]==1,x3[0]==1}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to -2 t-9 e^{-t}+7\\ \text {x2}(t)&\to e^{-t}\\ \text {x3}(t)&\to -t-2 e^{-t}+3 \end{align*}

✓ Sympy. Time used: 0.067 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(2*x__1(t) + x__2(t) - 4*x__3(t) + Derivative(x__1(t), t),0),Eq(x__2(t) + Derivative(x__2(t), t),0),Eq(x__1(t) + 3*x__2(t) - 2*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - 2 C_{1} - 2 C_{2} t + C_{2} + \frac {9 C_{3} e^{- t}}{2}, \ x^{2}{\left (t \right )} = - \frac {C_{3} e^{- t}}{2}, \ x^{3}{\left (t \right )} = - C_{1} - C_{2} t + C_{3} e^{- t}\right ] \]