20.15.5 problem 5

Internal problem ID [3831]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.3, page 598
Problem number : 5
Date solved : Monday, January 27, 2025 at 08:03:09 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 58

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

\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{4 t}+\frac {{\mathrm e}^{2 t} c_3}{3}-\frac {c_{1}}{3} \\ x_{2} \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{4 t}+{\mathrm e}^{2 t} c_3 \\ x_{3} \left (t \right ) &= c_{2} {\mathrm e}^{4 t}+\frac {{\mathrm e}^{2 t} c_3}{3}+\frac {5 c_{1}}{9} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 174

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

\begin{align*} \text {x1}(t)\to \frac {1}{8} \left (c_1 \left (2 e^{2 t}+3 e^{4 t}+3\right )-\left (e^{2 t}-1\right ) \left ((4 c_2-9 c_3) e^{2 t}-3 c_3\right )\right ) \\ \text {x2}(t)\to \frac {1}{8} \left (9 c_3 \left (e^{2 t}-1\right )^2+3 c_1 \left (2 e^{2 t}+e^{4 t}-3\right )-4 c_2 e^{2 t} \left (e^{2 t}-3\right )\right ) \\ \text {x3}(t)\to \frac {1}{8} \left (c_1 \left (2 e^{2 t}+3 e^{4 t}-5\right )+(4 c_2-6 c_3) e^{2 t}+(9 c_3-4 c_2) e^{4 t}+5 c_3\right ) \\ \end{align*}