20.20.19 problem 19

Internal problem ID [3909]
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 : 19
Date solved : Tuesday, March 04, 2025 at 05:19:15 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )-4 x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-4 x_{1} \left (t \right )-5 x_{2} \left (t \right )-6 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=4 x_{1} \left (t \right )+8 x_{2} \left (t \right )+7 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 101

ode:=[diff(x__1(t),t) = -x__1(t)-4*x__2(t)-2*x__3(t), diff(x__2(t),t) = -4*x__1(t)-5*x__2(t)-6*x__3(t), diff(x__3(t),t) = 4*x__1(t)+8*x__2(t)+7*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 172

ode={D[x1[t],t]==-1*x1[t]-4*x2[t]-2*x3[t],D[x2[t],t]==-4*x1[t]-5*x2[t]-6*x3[t],D[x3[t],t]==4*x1[t]+8*x2[t]+7*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.184 (sec). Leaf size: 105

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(x__1(t) + 4*x__2(t) + 2*x__3(t) + Derivative(x__1(t), t),0),Eq(4*x__1(t) + 5*x__2(t) + 6*x__3(t) + Derivative(x__2(t), t),0),Eq(-4*x__1(t) - 8*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)
 

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