20.17.9 problem 9

Internal problem ID [3863]
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 : 9
Date solved : Monday, January 27, 2025 at 08:03:34 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.070 (sec). Leaf size: 53

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 65

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

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