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 : Monday, January 27, 2025 at 09:23:14 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.239 (sec). Leaf size: 40

dsolve([diff(x(t),t)+x(t)+2*y(t)=8, 2*x(t)+diff(y(t),t)-2*y(t)=2*exp(-t)-8],singsol=all)
 

\begin{align*} x &= {\mathrm e}^{-2 t} c_{2} +c_{1} {\mathrm e}^{3 t}+{\mathrm e}^{-t} \\ y &= \frac {{\mathrm e}^{-2 t} c_{2}}{2}-2 c_{1} {\mathrm e}^{3 t}+4 \\ \end{align*}

✓ Solution by Mathematica

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

DSolve[{D[x[t],t]+x[t]+2*y[t]==8,2*x[t]+D[y[t],t]-2*y[t]==2*Exp[-t]-8},{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*}