14.29.2 problem 2

Internal problem ID [2800]
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 : 2
Date solved : Monday, January 27, 2025 at 06:13:58 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )-4 y\\ y^{\prime }&=2 x \left (t \right )+y \end{align*}

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 57

dsolve([diff(x(t),t)=-3*x(t)-4*y(t),diff(y(t),t)=2*x(t)+1*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 59

DSolve[{D[x[t],t]==-3*x[t]-4*y[t],D[y[t],t]==2*x[t]+1*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-t} (c_1 \cos (2 t)-(c_1+2 c_2) \sin (2 t)) \\ y(t)\to e^{-t} (c_2 \cos (2 t)+(c_1+c_2) \sin (2 t)) \\ \end{align*}