85.92.2 problem 1 (b)

Internal problem ID [23051]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 11. Matrix eigenvalue methods for systems of linear differential equations. A Exercises at page 528
Problem number : 1 (b)
Date solved : Thursday, October 02, 2025 at 09:18:21 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+x \left (t \right )-5 y \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )+4 x \left (t \right )+5 y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.074 (sec). Leaf size: 58

ode:=[diff(x(t),t)+x(t)-5*y(t) = 0, diff(y(t),t)+4*x(t)+5*y(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} \left (\sin \left (4 t \right ) c_1 +\cos \left (4 t \right ) c_2 \right ) \\ y \left (t \right ) &= -\frac {2 \,{\mathrm e}^{-3 t} \left (\sin \left (4 t \right ) c_1 +2 \sin \left (4 t \right ) c_2 -2 \cos \left (4 t \right ) c_1 +\cos \left (4 t \right ) c_2 \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 71

ode={D[x[t],{t,1}]+x[t]-5*y[t]==0, D[y[t],{t,1}]+4*x[t]+5*y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{4} e^{-3 t} (4 c_1 \cos (4 t)+(2 c_1+5 c_2) \sin (4 t))\\ y(t)&\to \frac {1}{2} e^{-3 t} (2 c_2 \cos (4 t)-(2 c_1+c_2) \sin (4 t)) \end{align*}

✓ Sympy. Time used: 0.070 (sec). Leaf size: 58

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) - 5*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)
 

\[ \left [ x{\left (t \right )} = \left (\frac {C_{1}}{2} + C_{2}\right ) e^{- 3 t} \sin {\left (4 t \right )} + \left (C_{1} - \frac {C_{2}}{2}\right ) e^{- 3 t} \cos {\left (4 t \right )}, \ y{\left (t \right )} = - C_{1} e^{- 3 t} \sin {\left (4 t \right )} + C_{2} e^{- 3 t} \cos {\left (4 t \right )}\right ] \]