14.32.1 problem 1

Internal problem ID [2825]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.7 (Phase portraits of linear systems). Page 427
Problem number : 1
Date solved : Monday, January 27, 2025 at 06:19:02 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=-5 x_{1} \left (t \right )+x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-5 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.028 (sec). Leaf size: 34

dsolve([diff(x__1(t),t)=-5*x__1(t)+x__2(t),diff(x__2(t),t)=1*x__1(t)-5*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-6 t}+c_2 \,{\mathrm e}^{-4 t} \\ x_{2} \left (t \right ) &= -c_1 \,{\mathrm e}^{-6 t}+c_2 \,{\mathrm e}^{-4 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 68

DSolve[{D[x1[t],t]==-5*x1[t]+x2[t],D[x2[t],t]==1*x1[t]-5*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-6 t} \left (c_1 \left (e^{2 t}+1\right )+c_2 \left (e^{2 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-6 t} \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^{2 t}+1\right )\right ) \\ \end{align*}