20.16.9 problem 9

Internal problem ID [3842]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.4 (Nondefective coefficient matrix), page 607
Problem number : 9
Date solved : Monday, January 27, 2025 at 08:03:17 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 53

dsolve([diff(x__1(t),t)=2*x__1(t)+3*x__3(t),diff(x__2(t),t)=-4*x__2(t),diff(x__3(t),t)=-3*x__1(t)+2*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 116

DSolve[{D[x1[t],t]==2*x1[t]+3*x3[t],D[x2[t],t]==-4*x2[t],D[x3[t],t]==-3*x1[t]+2*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{2 t} (c_1 \cos (3 t)+c_2 \sin (3 t)) \\ \text {x3}(t)\to e^{2 t} (c_2 \cos (3 t)-c_1 \sin (3 t)) \\ \text {x2}(t)\to c_3 e^{-4 t} \\ \text {x1}(t)\to e^{2 t} (c_1 \cos (3 t)+c_2 \sin (3 t)) \\ \text {x3}(t)\to e^{2 t} (c_2 \cos (3 t)-c_1 \sin (3 t)) \\ \text {x2}(t)\to 0 \\ \end{align*}