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 : Monday, January 27, 2025 at 08:04:22 AM
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*}

✓ Solution by Maple

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

dsolve([diff(x__1(t),t)=-1*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)],singsol=all)
 

\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*}

✓ Solution by Mathematica

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

DSolve[{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]},{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*}