14.20.13 problem 13

Internal problem ID [3903]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 13
Date solved : Tuesday, September 30, 2025 at 06:58:51 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-2 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-4 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{1} \left (t \right )+2 x_{2} \left (t \right )-3 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.180 (sec). Leaf size: 66

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

\begin{align*} x_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{2 t}+c_3 \,{\mathrm e}^{-5 t} \\ x_{2} \left (t \right ) &= \frac {c_2 \,{\mathrm e}^{2 t}}{4}+{\mathrm e}^{-2 t} c_1 +2 c_3 \,{\mathrm e}^{-5 t} \\ x_{3} \left (t \right ) &= \frac {c_2 \,{\mathrm e}^{2 t}}{2}+2 \,{\mathrm e}^{-2 t} c_1 -3 c_3 \,{\mathrm e}^{-5 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 153

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

\begin{align*} \text {x1}(t)&\to \frac {1}{7} e^{-5 t} \left ((7 c_1-2 c_2+c_3) e^{7 t}+2 c_2-c_3\right )\\ \text {x2}(t)&\to \frac {1}{28} e^{-5 t} \left (-7 (c_1-2 c_2-c_3) e^{3 t}+(7 c_1-2 c_2+c_3) e^{7 t}+16 c_2-8 c_3\right )\\ \text {x3}(t)&\to \frac {1}{14} e^{-5 t} \left (-7 (c_1-2 c_2-c_3) e^{3 t}+(7 c_1-2 c_2+c_3) e^{7 t}+6 (c_3-2 c_2)\right ) \end{align*}

✓ Sympy. Time used: 0.100 (sec). Leaf size: 70

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

\[ \left [ x^{1}{\left (t \right )} = - \frac {C_{1} e^{- 5 t}}{3} + 2 C_{2} e^{2 t}, \ x^{2}{\left (t \right )} = - \frac {2 C_{1} e^{- 5 t}}{3} + \frac {C_{2} e^{2 t}}{2} + \frac {C_{3} e^{- 2 t}}{2}, \ x^{3}{\left (t \right )} = C_{1} e^{- 5 t} + C_{2} e^{2 t} + C_{3} e^{- 2 t}\right ] \]