85.86.2 problem 1 (b)

Internal problem ID [23019]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. A Exercises at page 491
Problem number : 1 (b)
Date solved : Thursday, October 02, 2025 at 09:17:33 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*}

With initial conditions

\begin{align*} x \left (0\right )&=-1 \\ y \left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.080 (sec). Leaf size: 35

ode:=[diff(x(t),t)+x(t)-5*y(t) = 0, diff(y(t),t)+4*x(t)+5*y(t) = 0]; 
ic:=[x(0) = -1, y(0) = 2]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 37

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={x[0]==-1,y[0]==2}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -e^{-3 t} (\cos (4 t)-2 \sin (4 t))\\ y(t)&\to 2 e^{-3 t} \cos (4 t) \end{align*}

✓ Sympy. Time used: 0.085 (sec). Leaf size: 37

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 = {x(0): -1, y(0): 2} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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