14.29.3 problem 3

Internal problem ID [2801]
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 : 3
Date solved : Monday, January 27, 2025 at 06:13:59 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-5 x \left (t \right )+3 y\\ y^{\prime }&=-x \left (t \right )+y \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 80

dsolve([diff(x(t),t)=-5*x(t)+3*y(t),diff(y(t),t)=-1*x(t)+1*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\left (-2+\sqrt {6}\right ) t}+c_2 \,{\mathrm e}^{-\left (2+\sqrt {6}\right ) t} \\ y &= \frac {c_1 \,{\mathrm e}^{\left (-2+\sqrt {6}\right ) t} \sqrt {6}}{3}-\frac {c_2 \,{\mathrm e}^{-\left (2+\sqrt {6}\right ) t} \sqrt {6}}{3}+c_1 \,{\mathrm e}^{\left (-2+\sqrt {6}\right ) t}+c_2 \,{\mathrm e}^{-\left (2+\sqrt {6}\right ) t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 143

DSolve[{D[x[t],t]==-5*x[t]+3*y[t],D[y[t],t]==-1*x[t]+1*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{4} e^{-\left (\left (2+\sqrt {6}\right ) t\right )} \left (c_1 \left (-\left (\sqrt {6}-2\right ) e^{2 \sqrt {6} t}+2+\sqrt {6}\right )+\sqrt {6} c_2 \left (e^{2 \sqrt {6} t}-1\right )\right ) \\ y(t)\to \frac {1}{12} e^{-\left (\left (2+\sqrt {6}\right ) t\right )} \left (3 c_2 \left (\left (2+\sqrt {6}\right ) e^{2 \sqrt {6} t}+2-\sqrt {6}\right )-\sqrt {6} c_1 \left (e^{2 \sqrt {6} t}-1\right )\right ) \\ \end{align*}