65.16.2 problem 29.3 (ii)

Internal problem ID [13760]
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 : Wednesday, March 05, 2025 at 10:15:03 PM
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*}

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

ode:=[diff(x(t),t) = -2*x(t)+3*y(t), diff(y(t),t) = -6*x(t)+4*y(t)]; 
dsolve(ode);
 

\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*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 56

ode={D[x[t],t]==-2*x[t]+3*y[t],D[y[t],t]==-6*x[t]+4*y[t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.107 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(2*x(t) - 3*y(t) + Derivative(x(t), t),0),Eq(6*x(t) - 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{t} \sin {\left (3 t \right )} + \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{t} \cos {\left (3 t \right )}, \ y{\left (t \right )} = - C_{1} e^{t} \sin {\left (3 t \right )} + C_{2} e^{t} \cos {\left (3 t \right )}\right ] \]