85.88.11 problem 2 (a)

Internal problem ID [23038]
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 : 2 (a)
Date solved : Thursday, October 02, 2025 at 09:17:42 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+x \left (t \right )+2 y \left (t \right )&=8\\ 2 x \left (t \right )+\frac {d}{d t}y \left (t \right )-2 y \left (t \right )&=2 \,{\mathrm e}^{-t}-8 \end{align*}

✓ Maple. Time used: 0.068 (sec). Leaf size: 40

ode:=[diff(x(t),t)+x(t)+2*y(t) = 8, 2*x(t)+diff(y(t),t)-2*y(t) = 2*exp(-t)-8]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-2 t} c_2 +{\mathrm e}^{3 t} c_1 +{\mathrm e}^{-t} \\ y \left (t \right ) &= \frac {{\mathrm e}^{-2 t} c_2}{2}-2 \,{\mathrm e}^{3 t} c_1 +4 \\ \end{align*}

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 82

ode={D[x[t],{t,1}]+x[t]+2*y[t]==8, 2*x[t]+ D[y[t],{t,1}]-2*y[t]==2*Exp[-t]-8}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{5} e^{-2 t} \left (5 e^t+(c_1-2 c_2) e^{5 t}+2 (2 c_1+c_2)\right )\\ y(t)&\to \frac {1}{5} e^{-2 t} \left (20 e^{2 t}-2 (c_1-2 c_2) e^{5 t}+2 c_1+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.156 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) + 2*y(t) + Derivative(x(t), t) - 8,0),Eq(2*x(t) - 2*y(t) + Derivative(y(t), t) + 8 - 2*exp(-t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 2 C_{1} e^{- 2 t} - \frac {C_{2} e^{3 t}}{2} + e^{- t}, \ y{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{3 t} + 4\right ] \]