20.17.11 problem 11

Internal problem ID [3865]
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 : 11
Date solved : Monday, January 27, 2025 at 08:03:36 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 66

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 117

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

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