14.32.6 problem 6

Internal problem ID [2830]
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 : 6
Date solved : Monday, January 27, 2025 at 06:19:06 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=5 x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 34

dsolve([diff(x__1(t),t)=3*x__1(t)-x__2(t),diff(x__2(t),t)=5*x__1(t)-3*x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 73

DSolve[{D[x1[t],t]==3*x1[t]-x2[t],D[x2[t],t]==5*x1[t]-3*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

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