85.88.1 problem 1 (a)

Internal problem ID [23028]
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 (a)
Date solved : Thursday, October 02, 2025 at 09:17:36 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+6 x \left (t \right )+3 \frac {d}{d t}y \left (t \right )+2 y \left (t \right )&=0\\ \frac {d}{d t}x \left (t \right )+5 x \left (t \right )+2 \frac {d}{d t}y \left (t \right )+3 y \left (t \right )&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=2 \\ y \left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 31

ode:=[diff(x(t),t)+6*x(t)+3*diff(y(t),t)+2*y(t) = 0, diff(x(t),t)+5*x(t)+2*diff(y(t),t)+3*y(t) = 0]; 
ic:=[x(0) = 2, y(0) = 4]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 36

ode={D[x[t],{t,1}]+6*x[t]+3*D[y[t],t]+2*y[t]==0, D[x[t],{t,1}]+5*x[t]+2* D[y[t],{t,1}]+3*y[t]==0}; 
ic={x[0]==2,y[0]==4}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-4 t} \left (5-3 e^{6 t}\right )\\ y(t)&\to e^{-4 t}+3 e^{2 t} \end{align*}

✓ Sympy. Time used: 0.087 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(6*x(t) + 2*y(t) + Derivative(x(t), t) + 3*Derivative(y(t), t),0),Eq(5*x(t) + 3*y(t) + Derivative(x(t), t) + 2*Derivative(y(t), t),0)] 
ics = {x(0): 2, y(0): 4} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 3 e^{2 t} + 5 e^{- 4 t}, \ y{\left (t \right )} = 3 e^{2 t} + e^{- 4 t}\right ] \]