65.16.4 problem 29.3 (iv)

Internal problem ID [13762]
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 (iv)
Date solved : Wednesday, March 05, 2025 at 10:15:05 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=7 x \left (t \right )-5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=10 x \left (t \right )-3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 56

ode:=[diff(x(t),t) = 7*x(t)-5*y(t), diff(y(t),t) = 10*x(t)-3*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 62

ode={D[x[t],t]==7*x[t]-5*y[t],D[y[t],t]==10*x[t]-3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.113 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-7*x(t) + 5*y(t) + Derivative(x(t), t),0),Eq(-10*x(t) + 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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