76.6.10 problem 10

Internal problem ID [17465]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 10
Date solved : Tuesday, January 28, 2025 at 10:39:06 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )-y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.065 (sec). Leaf size: 44

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (c_{2} \cos \left (2 t \right )+c_{1} \sin \left (2 t \right )\right ) \\ y \left (t \right ) &= {\mathrm e}^{-t} \left (-\sin \left (2 t \right ) c_{2} +\cos \left (2 t \right ) c_{1} \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 51

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

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