20.17.1 problem 1

Internal problem ID [3855]
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 : 1
Date solved : Monday, January 27, 2025 at 08:03:28 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 )&=-x_{1} \left (t \right )+3 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 44

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

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