28.4.4 problem 7.4

Internal problem ID [4536]
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 : 7.4
Date solved : Tuesday, January 28, 2025 at 02:39:19 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime \prime }\left (t \right )+x^{\prime }\left (t \right )+y^{\prime }-2 y&=0\\ x^{\prime }\left (t \right )+x \left (t \right )-y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 37

dsolve([diff(x(t),t$2)+diff(x(t),t)+diff(y(t),t)-2*y(t)=0,diff(x(t),t)+x(t)-diff(y(t),t)=0],singsol=all)
 

\begin{align*} x &= c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{t}+c_3 \,{\mathrm e}^{-t} \\ y &= \frac {c_{1} {\mathrm e}^{-2 t}}{2}+2 c_{2} {\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 106

DSolve[{D[x[t],{t,2}]+D[x[t],t]+D[y[t],t]-2*y[t]==0,D[x[t],t]+x[t]-D[y[t],t]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{6} e^{-2 t} \left (2 c_3 \left (e^t+2\right ) \left (e^t-1\right )^2+c_1 \left (9 e^t+e^{3 t}-4\right )+c_2 \left (3 e^t+e^{3 t}-4\right )\right ) \\ y(t)\to \frac {1}{3} e^{-2 t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}-1\right )+c_3 \left (2 e^{3 t}+1\right )\right ) \\ \end{align*}