20.13.3 problem 3

Internal problem ID [3812]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.1, page 587
Problem number : 3
Date solved : Monday, January 27, 2025 at 08:02:54 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 58

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

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