14.29.15 problem 16

Internal problem ID [2813]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.2 (Stability of linear systems). Page 383
Problem number : 16
Date solved : Monday, January 27, 2025 at 06:14:12 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 60

dsolve([diff(x__1(t),t)=-x__1(t)-x__2(t)+1,diff(x__2(t),t)=2*x__1(t)-x__2(t)+5],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {4}{3}+{\mathrm e}^{-t} \left (\cos \left (\sqrt {2}\, t \right ) c_1 +c_2 \sin \left (\sqrt {2}\, t \right )\right ) \\ x_{2} \left (t \right ) &= \frac {7}{3}-{\mathrm e}^{-t} \sqrt {2}\, \left (c_2 \cos \left (\sqrt {2}\, t \right )-c_1 \sin \left (\sqrt {2}\, t \right )\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.070 (sec). Leaf size: 91

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

\begin{align*} \text {x1}(t)\to c_1 e^{-t} \cos \left (\sqrt {2} t\right )-\frac {c_2 e^{-t} \sin \left (\sqrt {2} t\right )}{\sqrt {2}}-\frac {5}{3} \\ \text {x2}(t)\to c_2 e^{-t} \cos \left (\sqrt {2} t\right )+\sqrt {2} c_1 e^{-t} \sin \left (\sqrt {2} t\right )+\frac {5}{3} \\ \end{align*}