20.20.14 problem 14

Internal problem ID [3904]
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 : 14
Date solved : Monday, January 27, 2025 at 08:04:17 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-4 x_{2} \left (t \right )+3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-9 x_{1} \left (t \right )-3 x_{2} \left (t \right )-9 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=4 x_{1} \left (t \right )+4 x_{2} \left (t \right )+3 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 59

dsolve([diff(x__1(t),t)=2*x__1(t)-4*x__2(t)+3*x__3(t),diff(x__2(t),t)=-9*x__1(t)-3*x__2(t)-9*x__3(t),diff(x__3(t),t)=4*x__1(t)+4*x__2(t)+3*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{6 t}+c_3 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= \frac {c_{1} {\mathrm e}^{-3 t}}{2}-c_{2} {\mathrm e}^{6 t} \\ x_{3} \left (t \right ) &= -c_{1} {\mathrm e}^{-3 t}-c_3 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 135

DSolve[{D[x1[t],t]==2*x1[t]-4*x2[t]+3*x3[t],D[x2[t],t]==-9*x1[t]-3*x2[t]-9*x3[t],D[x3[t],t]==4*x1[t]+4*x2[t]+3*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-3 t} \left (c_1 \left (-2 e^{2 t}+e^{9 t}+2\right )-2 c_2 \left (e^{2 t}-1\right )+c_3 \left (-3 e^{2 t}+e^{9 t}+2\right )\right ) \\ \text {x2}(t)\to e^{-3 t} \left (c_1 \left (-e^{9 t}\right )-c_3 e^{9 t}+c_1+c_2+c_3\right ) \\ \text {x3}(t)\to e^{-3 t} \left (2 c_1 \left (e^{2 t}-1\right )+2 c_2 \left (e^{2 t}-1\right )+c_3 \left (3 e^{2 t}-2\right )\right ) \\ \end{align*}