14.32.9 problem 9

Internal problem ID [2833]
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 : 9
Date solved : Monday, January 27, 2025 at 06:19:08 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 36

dsolve([diff(x__1(t),t)=2*x__1(t)+x__2(t),diff(x__2(t),t)=-5*x__1(t)-2*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \sin \left (t \right )+c_2 \cos \left (t \right ) \\ x_{2} \left (t \right ) &= \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 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 41

DSolve[{D[x1[t],t]==2*x1[t]+x2[t],D[x2[t],t]==-5*x1[t]-2*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 \cos (t)+(2 c_1+c_2) \sin (t) \\ \text {x2}(t)\to c_2 (\cos (t)-2 \sin (t))-5 c_1 \sin (t) \\ \end{align*}