85.88.5 problem 1 (e)

Internal problem ID [23032]
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 499
Problem number : 1 (e)
Date solved : Thursday, October 02, 2025 at 09:17:38 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+3 x \left (t \right )+2 y \left (t \right )&=0\\ 3 x \left (t \right )+\frac {d}{d t}y \left (t \right )+y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 82

ode:=[diff(x(t),t)+3*x(t)+2*y(t) = 0, 3*x(t)+diff(y(t),t)+y(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\left (-2+\sqrt {7}\right ) t}+c_2 \,{\mathrm e}^{-\left (2+\sqrt {7}\right ) t} \\ y \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{\left (-2+\sqrt {7}\right ) t} \sqrt {7}}{2}+\frac {c_2 \,{\mathrm e}^{-\left (2+\sqrt {7}\right ) t} \sqrt {7}}{2}-\frac {c_1 \,{\mathrm e}^{\left (-2+\sqrt {7}\right ) t}}{2}-\frac {c_2 \,{\mathrm e}^{-\left (2+\sqrt {7}\right ) t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 143

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

\begin{align*} x(t)&\to \frac {1}{14} e^{-\left (\left (2+\sqrt {7}\right ) t\right )} \left (c_1 \left (-\left (\sqrt {7}-7\right ) e^{2 \sqrt {7} t}+7+\sqrt {7}\right )-2 \sqrt {7} c_2 \left (e^{2 \sqrt {7} t}-1\right )\right )\\ y(t)&\to \frac {1}{14} e^{-\left (\left (2+\sqrt {7}\right ) t\right )} \left (c_2 \left (\left (7+\sqrt {7}\right ) e^{2 \sqrt {7} t}+7-\sqrt {7}\right )-3 \sqrt {7} c_1 \left (e^{2 \sqrt {7} t}-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.133 (sec). Leaf size: 68

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

\[ \left [ x{\left (t \right )} = \frac {C_{1} \left (1 - \sqrt {7}\right ) e^{- t \left (2 - \sqrt {7}\right )}}{3} + \frac {C_{2} \left (1 + \sqrt {7}\right ) e^{- t \left (2 + \sqrt {7}\right )}}{3}, \ y{\left (t \right )} = C_{1} e^{- t \left (2 - \sqrt {7}\right )} + C_{2} e^{- t \left (2 + \sqrt {7}\right )}\right ] \]