20.20.17 problem 17

Internal problem ID [3907]
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 : 17
Date solved : Tuesday, March 04, 2025 at 05:19:13 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-7 x_{1} \left (t \right )-6 x_{2} \left (t \right )-7 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-3 x_{1} \left (t \right )-3 x_{2} \left (t \right )-3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=7 x_{1} \left (t \right )+6 x_{2} \left (t \right )+7 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 48

ode:=[diff(x__1(t),t) = -7*x__1(t)-6*x__2(t)-7*x__3(t), diff(x__2(t),t) = -3*x__1(t)-3*x__2(t)-3*x__3(t), diff(x__3(t),t) = 7*x__1(t)+6*x__2(t)+7*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 136

ode={D[x1[t],t]==-7*x1[t]-6*x2[t]-7*x3[t],D[x2[t],t]==-3*x1[t]-3*x2[t]-3*x3[t],D[x3[t],t]==7*x1[t]+6*x2[t]+7*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-3 t} \left (-\left (c_1 \left (e^{3 t} (t+1)-2\right )\right )-2 c_2 \left (e^{3 t}-1\right )-c_3 \left (e^{3 t} (t+2)-2\right )\right ) \\ \text {x2}(t)\to e^{-3 t} \left (c_1 \left (-e^{3 t}\right )-c_3 e^{3 t}+c_1+c_2+c_3\right ) \\ \text {x3}(t)\to e^{-3 t} \left (c_1 \left (e^{3 t} (t+2)-2\right )+2 c_2 \left (e^{3 t}-1\right )+c_3 \left (e^{3 t} (t+3)-2\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.120 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(7*x__1(t) + 6*x__2(t) + 7*x__3(t) + Derivative(x__1(t), t),0),Eq(3*x__1(t) + 3*x__2(t) + 3*x__3(t) + Derivative(x__2(t), t),0),Eq(-7*x__1(t) - 6*x__2(t) - 7*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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