71.19.5 problem 5

Internal problem ID [14597]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 10. Applications of Systems of Equations. Exercises 10.2 page 432
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 06:44:04 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: 0.043 (sec). Leaf size: 44

dsolve([diff(x(t),t)=-x(t)+2*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 \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]-1*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*}