14.32.2 problem 2

Internal problem ID [2826]
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 : 2
Date solved : Monday, January 27, 2025 at 06:19:03 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=8 x_{1} \left (t \right )-6 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

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

dsolve([diff(x__1(t),t)=0*x__1(t)-x__2(t),diff(x__2(t),t)=8*x__1(t)-6*x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 70

DSolve[{D[x1[t],t]==0*x1[t]-x2[t],D[x2[t],t]==8*x1[t]-6*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

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