65.16.3 problem 29.3 (iii)

Internal problem ID [13840]
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 (iii)
Date solved : Tuesday, January 28, 2025 at 06:04:58 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-11 x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=13 x \left (t \right )-9 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 58

dsolve([diff(x(t),t)=-11*x(t)-2*y(t),diff(y(t),t)=13*x(t)-9*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-10 t} \left (\sin \left (5 t \right ) c_{1} +\cos \left (5 t \right ) c_{2} \right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-10 t} \left (\sin \left (5 t \right ) c_{1} -5 \sin \left (5 t \right ) c_{2} +5 \cos \left (5 t \right ) c_{1} +\cos \left (5 t \right ) c_{2} \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 69

DSolve[{D[x[t],t]==-11*x[t]-2*y[t],D[y[t],t]==13*x[t]-9*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{5} e^{-10 t} (5 c_1 \cos (5 t)-(c_1+2 c_2) \sin (5 t)) \\ y(t)\to \frac {1}{5} e^{-10 t} (5 c_2 \cos (5 t)+(13 c_1+c_2) \sin (5 t)) \\ \end{align*}