65.16.3 problem 29.3 (iii)

Internal problem ID [13761]
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 : Wednesday, March 05, 2025 at 10:15:04 PM
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*}

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

ode:=[diff(x(t),t) = -11*x(t)-2*y(t), diff(y(t),t) = 13*x(t)-9*y(t)]; 
dsolve(ode);
 

\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*}

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

ode={D[x[t],t]==-11*x[t]-2*y[t],D[y[t],t]==13*x[t]-9*y[t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.116 (sec). Leaf size: 66

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(11*x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-13*x(t) + 9*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {C_{1}}{13} + \frac {5 C_{2}}{13}\right ) e^{- 10 t} \cos {\left (5 t \right )} - \left (\frac {5 C_{1}}{13} - \frac {C_{2}}{13}\right ) e^{- 10 t} \sin {\left (5 t \right )}, \ y{\left (t \right )} = C_{1} e^{- 10 t} \cos {\left (5 t \right )} - C_{2} e^{- 10 t} \sin {\left (5 t \right )}\right ] \]