65.16.2 problem 29.3 (ii)

Internal problem ID [13839]
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 (ii)
Date solved : Tuesday, January 28, 2025 at 06:04:57 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-2 x \left (t \right )+3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-6 x \left (t \right )+4 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 52

dsolve([diff(x(t),t)=-2*x(t)+3*y(t),diff(y(t),t)=-6*x(t)+4*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 56

DSolve[{D[x[t],t]==-2*x[t]+3*y[t],D[y[t],t]==-6*x[t]+4*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^t (c_1 \cos (3 t)+(c_2-c_1) \sin (3 t)) \\ y(t)\to e^t (c_2 \cos (3 t)+(c_2-2 c_1) \sin (3 t)) \\ \end{align*}