20.13.8 problem 8

Internal problem ID [3817]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.1, page 587
Problem number : 8
Date solved : Monday, January 27, 2025 at 08:02:58 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.051 (sec). Leaf size: 68

dsolve([diff(x__1(t),t)=-2*x__1(t)+x__2(t)+x__3(t),diff(x__2(t),t)=x__1(t)-x__2(t)+3*x__3(t),diff(x__3(t),t)=-x__2(t)-3*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 114

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

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