20.17.16 problem 16

Internal problem ID [3870]
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 : 16
Date solved : Monday, January 27, 2025 at 08:03:40 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = -2\\ x_{2} \left (0\right ) = 1\\ x_{3} \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 36

dsolve([diff(x__1(t),t) = -2*x__1(t)-x__2(t)+4*x__3(t), diff(x__2(t),t) = -x__2(t), diff(x__3(t),t) = -x__1(t)-3*x__2(t)+2*x__3(t), x__1(0) = -2, x__2(0) = 1, x__3(0) = 1], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= -9 \,{\mathrm e}^{-t}-2 t +7 \\ x_{2} \left (t \right ) &= {\mathrm e}^{-t} \\ x_{3} \left (t \right ) &= -2 \,{\mathrm e}^{-t}+3-t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 40

DSolve[{D[x1[t],t]==-2*x1[t]-x2[t]+4*x3[t],D[x2[t],t]==-x2[t],D[x3[t],t]==-x1[t]-3*x2[t]+2*x3[t]},{x1[0]==-2,x2[0]==1,x3[0]==1},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to -2 t-9 e^{-t}+7 \\ \text {x2}(t)\to e^{-t} \\ \text {x3}(t)\to -t-2 e^{-t}+3 \\ \end{align*}