20.20.23 problem 23

Internal problem ID [3913]
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 : 23
Date solved : Monday, January 27, 2025 at 08:04:26 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.057 (sec). Leaf size: 72

dsolve([diff(x__1(t),t)=2*x__1(t)+13*x__2(t)-0*x__3(t)+0*x__4(t),diff(x__2(t),t)=-1*x__1(t)-2*x__2(t)-0*x__3(t)+0*x__4(t),diff(x__3(t),t)=0*x__1(t)+0*x__2(t)+2*x__3(t)+4*x__4(t),diff(x__4(t),t)=0*x__1(t)+0*x__2(t)+0*x__3(t)+2*x__4(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \sin \left (3 t \right ) c_{1} +\cos \left (3 t \right ) c_{2} \\ x_{2} \left (t \right ) &= \frac {3 \cos \left (3 t \right ) c_{1}}{13}-\frac {3 \sin \left (3 t \right ) c_{2}}{13}-\frac {2 \sin \left (3 t \right ) c_{1}}{13}-\frac {2 \cos \left (3 t \right ) c_{2}}{13} \\ x_{3} \left (t \right ) &= \left (4 c_4 t +c_3 \right ) {\mathrm e}^{2 t} \\ x_{4} \left (t \right ) &= c_4 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 84

DSolve[{D[x1[t],t]==2*x1[t]+13*x2[t]-0*x3[t]+0*x4[t],D[x2[t],t]==-1*x1[t]-2*x2[t]-0*x3[t]+0*x4[t],D[x3[t],t]==0*x1[t]+0*x2[t]+2*x3[t]+4*x4[t],D[x4[t],t]==0*x1[t]+0*x2[t]+0*x3[t]+2*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 \cos (3 t)+\frac {1}{3} (2 c_1+13 c_2) \sin (3 t) \\ \text {x2}(t)\to c_2 \cos (3 t)-\frac {1}{3} (c_1+2 c_2) \sin (3 t) \\ \text {x3}(t)\to e^{2 t} (4 c_4 t+c_3) \\ \text {x4}(t)\to c_4 e^{2 t} \\ \end{align*}