63.22.10 problem 6

Internal problem ID [13240]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 237
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 05:12:53 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 26

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

\begin{align*} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{5 t} \\ y &= -2 c_{2} {\mathrm e}^{5 t}+\frac {c_{1}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 62

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

\begin{align*} x(t)\to \frac {1}{5} \left (c_1 \left (e^{5 t}+4\right )-2 c_2 \left (e^{5 t}-1\right )\right ) \\ y(t)\to \frac {1}{5} \left (c_2 \left (4 e^{5 t}+1\right )-2 c_1 \left (e^{5 t}-1\right )\right ) \\ \end{align*}