20.16.1 problem 1

Internal problem ID [3834]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.4 (Nondefective coefficient matrix), page 607
Problem number : 1
Date solved : Monday, January 27, 2025 at 08:03:11 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 72

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

\begin{align*} \text {x1}(t)\to \frac {1}{5} e^{-2 t} \left (c_1 \left (e^{5 t}+4\right )+2 c_2 \left (e^{5 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{5} e^{-2 t} \left (2 c_1 \left (e^{5 t}-1\right )+c_2 \left (4 e^{5 t}+1\right )\right ) \\ \end{align*}