20.20.20 problem 20

Internal problem ID [3910]
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 : 20
Date solved : Monday, January 27, 2025 at 08:04:23 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=7 x_{1} \left (t \right )-2 x_{2} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{2} \left (t \right )-x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{1} \left (t \right )+x_{2} \left (t \right )+4 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 68

dsolve([diff(x__1(t),t)=7*x__1(t)-2*x__2(t)+2*x__3(t),diff(x__2(t),t)=0*x__1(t)+4*x__2(t)-1*x__3(t),diff(x__3(t),t)=-1*x__1(t)+1*x__2(t)+4*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 105

DSolve[{D[x1[t],t]==7*x1[t]-2*x2[t]+2*x3[t],D[x2[t],t]==0*x1[t]+4*x2[t]-1*x3[t],D[x3[t],t]==-1*x1[t]+1*x2[t]+4*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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