20.20.21 problem 21

Internal problem ID [3911]
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 : 21
Date solved : Tuesday, March 04, 2025 at 05:19:17 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 )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{1} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 63

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 98

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

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

✓ Sympy. Time used: 0.137 (sec). Leaf size: 73

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

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