63.18.1 problem 2(a)

Internal problem ID [13207]
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(a)
Date solved : Tuesday, January 28, 2025 at 05:12:26 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=-3 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \end{align*}

✓ Solution by Maple

Time used: 1.142 (sec). Leaf size: 47

dsolve([diff(x(t),t)=-3*y(t),diff(y(t),t)=2*x(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} \sin \left (\sqrt {6}\, t \right )+c_{2} \cos \left (\sqrt {6}\, t \right ) \\ y &= -\frac {\sqrt {6}\, \left (\cos \left (\sqrt {6}\, t \right ) c_{1} -\sin \left (\sqrt {6}\, t \right ) c_{2} \right )}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 69

DSolve[{D[x[t],t]==-3*y[t],D[y[t],t]==2*x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to c_1 \cos \left (\sqrt {6} t\right )-\sqrt {\frac {3}{2}} c_2 \sin \left (\sqrt {6} t\right ) \\ y(t)\to c_2 \cos \left (\sqrt {6} t\right )+\sqrt {\frac {2}{3}} c_1 \sin \left (\sqrt {6} t\right ) \\ \end{align*}