20.17.4 problem 4

Internal problem ID [3858]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.5 (Defective coefficient matrix), page 619
Problem number : 4
Date solved : Monday, January 27, 2025 at 08:03:31 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 81

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

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 152

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

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