14.32.4 problem 4

Internal problem ID [2828]
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 : 4
Date solved : Monday, January 27, 2025 at 06:19:04 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=-4 x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-6 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 1.309 (sec). Leaf size: 30

dsolve([diff(x__1(t),t)=-4*x__1(t)-x__2(t),diff(x__2(t),t)=1*x__1(t)-6*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-5 t} \left (c_2 t +c_1 \right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{-5 t} \left (c_2 t +c_1 -c_2 \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 44

DSolve[{D[x1[t],t]==-4*x1[t]-x2[t],D[x2[t],t]==1*x1[t]-6*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

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