14.32.8 problem 8

Internal problem ID [2832]
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 : 8
Date solved : Monday, January 27, 2025 at 06:19:08 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=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: 0.016 (sec). Leaf size: 47

dsolve([diff(x__1(t),t)=1*x__1(t)-1*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 ) &= {\mathrm e}^{-t} \left (c_1 \sin \left (t \right )+c_2 \cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= -{\mathrm e}^{-t} \left (\cos \left (t \right ) c_1 -2 c_2 \cos \left (t \right )-2 c_1 \sin \left (t \right )-\sin \left (t \right ) c_2 \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 56

DSolve[{D[x1[t],t]==1*x1[t]-1*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 e^{-t} (c_1 \cos (t)+(2 c_1-c_2) \sin (t)) \\ \text {x2}(t)\to e^{-t} (c_2 \cos (t)+(5 c_1-2 c_2) \sin (t)) \\ \end{align*}