Internal problem ID [12102]
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).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 82
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 (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} -\sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) 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[{x'[t]==-y[t],y'[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*}