20.17.8 problem 8

Internal problem ID [3862]
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 : 8
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 )&=4 x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{2} \left (t \right )+4 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.067 (sec). Leaf size: 45

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 57

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

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