78.29.7 problem 1 (g)

Internal problem ID [18407]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 10. Systems of First Order Equations. Section 60. Critical Points and Stability for Linear Systems. Problems at page 539
Problem number : 1 (g)
Date solved : Thursday, March 13, 2025 at 11:55:44 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=5 x \left (t \right )+2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 58

ode:=[diff(x(t),t) = 4*x(t)-2*y(t), diff(y(t),t) = 5*x(t)+2*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 70

ode={D[x[t],t]==4*x[t]-2*y[t],D[y[t],t]==5*x[t]+2*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{3 t} (3 c_1 \cos (3 t)+(c_1-2 c_2) \sin (3 t)) \\ y(t)\to \frac {1}{3} e^{3 t} (3 c_2 \cos (3 t)+(5 c_1-c_2) \sin (3 t)) \\ \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-5*x(t) - 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \left (\frac {C_{1}}{5} - \frac {3 C_{2}}{5}\right ) e^{3 t} \cos {\left (3 t \right )} - \left (\frac {3 C_{1}}{5} + \frac {C_{2}}{5}\right ) e^{3 t} \sin {\left (3 t \right )}, \ y{\left (t \right )} = C_{1} e^{3 t} \cos {\left (3 t \right )} - C_{2} e^{3 t} \sin {\left (3 t \right )}\right ] \]