20.16.8 problem 8

Internal problem ID [3841]
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 : 8
Date solved : Monday, January 27, 2025 at 08:03:16 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 35

dsolve([diff(x__1(t),t)=x__2(t),diff(x__2(t),t)=-x__1(t),diff(x__3(t),t)=5*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ x_{2} \left (t \right ) &= \cos \left (t \right ) c_{1} -\sin \left (t \right ) c_{2} \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{5 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 76

DSolve[{D[x1[t],t]==x2[t],D[x2[t],t]==-x1[t],D[x3[t],t]==5*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 \cos (t)+c_2 \sin (t) \\ \text {x2}(t)\to c_2 \cos (t)-c_1 \sin (t) \\ \text {x3}(t)\to c_3 e^{5 t} \\ \text {x1}(t)\to c_1 \cos (t)+c_2 \sin (t) \\ \text {x2}(t)\to c_2 \cos (t)-c_1 \sin (t) \\ \text {x3}(t)\to 0 \\ \end{align*}