65.16.1 problem 29.3 (i)

Internal problem ID [13838]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 29, Complex eigenvalues. Exercises page 292
Problem number : 29.3 (i)
Date solved : Tuesday, January 28, 2025 at 06:04:56 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 81

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

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

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 112

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

\begin{align*} x(t)\to \frac {1}{3} e^{-t/2} \left (3 c_1 \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} (c_1-2 c_2) \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \\ y(t)\to \frac {1}{3} e^{-t/2} \left (3 c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} (2 c_1-c_2) \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \\ \end{align*}