20.20.32 problem 36

Internal problem ID [3922]
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 : 36
Date solved : Monday, January 27, 2025 at 08:04:34 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=5 x_{1} \left (t \right )+9 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 59

dsolve([diff(x__1(t),t)=5*x__1(t)+9*x__2(t),diff(x__2(t),t)=-2*x__1(t)-1*x__2(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 ) &= \frac {{\mathrm e}^{2 t} \left (-\sin \left (3 t \right ) c_{1} -\sin \left (3 t \right ) c_{2} +\cos \left (3 t \right ) c_{1} -\cos \left (3 t \right ) c_{2} \right )}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 67

DSolve[{D[x1[t],t]==5*x1[t]+9*x2[t],D[x2[t],t]==-2*x1[t]-1*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{2 t} (c_1 \cos (3 t)+(c_1+3 c_2) \sin (3 t)) \\ \text {x2}(t)\to \frac {1}{3} e^{2 t} (3 c_2 \cos (3 t)-(2 c_1+3 c_2) \sin (3 t)) \\ \end{align*}