63.18.2 problem 2(b)

Internal problem ID [13208]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 190
Problem number : 2(b)
Date solved : Tuesday, January 28, 2025 at 05:12:27 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 51

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

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

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 111

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

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