73.28.5 problem 39.2

Internal problem ID [15780]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 39. Critical points, Direction fields and trajectories. Additional Exercises. page 815
Problem number : 39.2
Date solved : Tuesday, January 28, 2025 at 08:07:03 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: 0.029 (sec). Leaf size: 30

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 ) &= c_{1} {\mathrm e}^{t}+{\mathrm e}^{-3 t} c_{2} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}-{\mathrm e}^{-3 t} c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 68

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