85.92.1 problem 1 (a)

Internal problem ID [23050]
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 (a)
Date solved : Thursday, October 02, 2025 at 09:18:20 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+5 x \left (t \right )-4 y \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )-x \left (t \right )+2 y \left (t \right )&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=3 \\ y \left (0\right )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 33

ode:=[diff(x(t),t)+5*x(t)-4*y(t) = 0, diff(y(t),t)-x(t)+2*y(t) = 0]; 
ic:=[x(0) = 3, y(0) = -2]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= 4 \,{\mathrm e}^{-6 t}-{\mathrm e}^{-t} \\ y \left (t \right ) &= -{\mathrm e}^{-6 t}-{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 36

ode={D[x[t],{t,1}]+5*x[t]-4*y[t]==0, D[y[t],{t,1}]-x[t]+2*y[t]==0}; 
ic={x[0]==3,y[0]==-2}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -e^{-6 t} \left (e^{5 t}-4\right )\\ y(t)&\to -e^{-6 t} \left (e^{5 t}+1\right ) \end{align*}

✓ Sympy. Time used: 0.059 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(5*x(t) - 4*y(t) + Derivative(x(t), t),0),Eq(-x(t) + 2*y(t) + Derivative(y(t), t),0)] 
ics = {x(0): 3, y(0): -2} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - e^{- t} + 4 e^{- 6 t}, \ y{\left (t \right )} = - e^{- t} - e^{- 6 t}\right ] \]