20.20.9 problem 9

Internal problem ID [3899]
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 : 9
Date solved : Monday, January 27, 2025 at 08:04:13 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )+13 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 49

dsolve([diff(x__1(t),t)=3*x__1(t)+13*x__2(t),diff(x__2(t),t)=-x__1(t)-3*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right ) \\ x_{2} \left (t \right ) &= \frac {2 c_{1} \cos \left (2 t \right )}{13}-\frac {2 c_{2} \sin \left (2 t \right )}{13}-\frac {3 c_{1} \sin \left (2 t \right )}{13}-\frac {3 c_{2} \cos \left (2 t \right )}{13} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 51

DSolve[{D[x1[t],t]==3*x1[t]+13*x2[t],D[x2[t],t]==-1*x1[t]-3*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 \cos (2 t)+(3 c_1+13 c_2) \sin (t) \cos (t) \\ \text {x2}(t)\to c_2 \cos (2 t)-(c_1+3 c_2) \sin (t) \cos (t) \\ \end{align*}