78.29.2 problem 1 (b)

Internal problem ID [18410]
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 (b)
Date solved : Monday, March 31, 2025 at 05:27:36 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.131 (sec). Leaf size: 56

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} \left (\sin \left (2 t \right ) c_1 +\cos \left (2 t \right ) c_2 \right ) \\ y \left (t \right ) &= {\mathrm e}^{-3 t} \left (\sin \left (2 t \right ) c_1 +\sin \left (2 t \right ) c_2 -\cos \left (2 t \right ) c_1 +\cos \left (2 t \right ) c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 62

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

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

✓ Sympy. Time used: 0.115 (sec). Leaf size: 61

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

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