20.17.5 problem 5

Internal problem ID [3859]
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 : 5
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 )&=2 x_{1} \left (t \right )+2 x_{2} \left (t \right )-x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{1} \left (t \right )+3 x_{2} \left (t \right )-x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 46

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

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 138

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

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