20.20.8 problem 8

Internal problem ID [3898]
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 : 8
Date solved : Tuesday, March 04, 2025 at 05:19:02 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )-7 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=6 x_{1} \left (t \right )+6 x_{2} \left (t \right )+4 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.066 (sec). Leaf size: 60

ode:=[diff(x__1(t),t) = -3*x__1(t)-x__2(t), diff(x__2(t),t) = 4*x__1(t)-7*x__2(t), diff(x__3(t),t) = 6*x__1(t)+6*x__2(t)+4*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-5 t} \left (c_3 t +c_{2} \right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{-5 t} \left (2 c_3 t -c_3 +2 c_{2} \right ) \\ x_{3} \left (t \right ) &= \left (-\frac {2 \,{\mathrm e}^{-9 t} \left (9 c_3 t -2 c_3 +9 c_{2} \right )}{9}+c_{1} \right ) {\mathrm e}^{4 t} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 100

ode={D[x1[t],t]==-3*x1[t]-1*x2[t]+0*x3[t],D[x2[t],t]==4*x1[t]-7*x2[t]+0*x3[t],D[x3[t],t]==6*x1[t]+6*x2[t]+4*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-5 t} (2 c_1 t-c_2 t+c_1) \\ \text {x2}(t)\to e^{-5 t} (4 c_1 t-2 c_2 t+c_2) \\ \text {x3}(t)\to \frac {1}{9} e^{-5 t} \left (2 c_1 \left (-18 t+5 e^{9 t}-5\right )+2 c_2 \left (9 t+2 e^{9 t}-2\right )+9 c_3 e^{9 t}\right ) \\ \end{align*}

✓ Sympy. Time used: 0.173 (sec). Leaf size: 88

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(3*x__1(t) + x__2(t) + Derivative(x__1(t), t),0),Eq(-4*x__1(t) + 7*x__2(t) + Derivative(x__2(t), t),0),Eq(-6*x__1(t) - 6*x__2(t) - 4*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {9 C_{2} t e^{- 5 t}}{5} - \left (\frac {9 C_{1}}{5} + \frac {2 C_{2}}{5}\right ) e^{- 5 t}, \ x^{2}{\left (t \right )} = - \frac {18 C_{2} t e^{- 5 t}}{5} - \left (\frac {18 C_{1}}{5} - C_{2}\right ) e^{- 5 t}, \ x^{3}{\left (t \right )} = \frac {18 C_{1} e^{- 5 t}}{5} + \frac {18 C_{2} t e^{- 5 t}}{5} + C_{3} e^{4 t}\right ] \]