78.5.8 problem 5.b

Internal problem ID [21047]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 6, Linear systems. Problems section 6.9
Problem number : 5.b
Date solved : Thursday, October 02, 2025 at 07:01:45 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-y\\ y^{\prime }&=-5 x \left (t \right ) \end{align*}

✓ Maple. Time used: 0.099 (sec). Leaf size: 49

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\sqrt {5}\, t}+c_2 \,{\mathrm e}^{-\sqrt {5}\, t} \\ y \left (t \right ) &= -\sqrt {5}\, \left (c_1 \,{\mathrm e}^{\sqrt {5}\, t}-c_2 \,{\mathrm e}^{-\sqrt {5}\, t}\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 111

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

\begin{align*} x(t)&\to \frac {1}{10} e^{-\sqrt {5} t} \left (5 c_1 \left (e^{2 \sqrt {5} t}+1\right )-\sqrt {5} c_2 \left (e^{2 \sqrt {5} t}-1\right )\right )\\ y(t)&\to \frac {1}{2} e^{-\sqrt {5} t} \left (c_2 \left (e^{2 \sqrt {5} t}+1\right )-\sqrt {5} c_1 \left (e^{2 \sqrt {5} t}-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.097 (sec). Leaf size: 58

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

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