20.20.18 problem 18

Internal problem ID [3908]
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 : 18
Date solved : Monday, January 27, 2025 at 08:04:22 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )-x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+6 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{1} \left (t \right )+6 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 63

dsolve([diff(x__1(t),t)=3*x__1(t)-1*x__2(t)-2*x__3(t),diff(x__2(t),t)=1*x__1(t)+6*x__2(t)+1*x__3(t),diff(x__3(t),t)=1*x__1(t)+0*x__2(t)+6*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 98

DSolve[{D[x1[t],t]==3*x1[t]-1*x2[t]-2*x3[t],D[x2[t],t]==1*x1[t]+6*x2[t]+1*x3[t],D[x3[t],t]==1*x1[t]+0*x2[t]+6*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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