28.4.8 problem 7.8

Internal problem ID [4540]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.8
Date solved : Sunday, March 30, 2025 at 03:25:48 AM
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: 0.164 (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}^{3 t} c_2 +{\mathrm e}^{-2 t} c_1 +{\mathrm e}^{-t} \\ y \left (t \right ) &= -2 \,{\mathrm e}^{3 t} c_2 +\frac {{\mathrm e}^{-2 t} c_1}{2}+4 \\ \end{align*}

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 82

ode={D[x[t],t]+x[t]+2*y[t]==8,2*x[t]+D[y[t],t]-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: 0.231 (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)
 

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