28.4.30 problem 7.30

Internal problem ID [4562]
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 : 7.30
Date solved : Monday, January 27, 2025 at 09:23:31 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-4 x_{1} \left (t \right )+x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.028 (sec). Leaf size: 35

dsolve([diff(x__1(t),t)=x__1(t)-x__2(t),diff(x__2(t),t)=-4*x__1(t)+x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= -2 c_{1} {\mathrm e}^{3 t}+2 c_{2} {\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 71

DSolve[{D[x1[t],t]==x1[t]-x2[t],D[x2[t],t]==-4*x1[t]+x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-t} \left (2 c_1 \left (e^{4 t}+1\right )-c_2 \left (e^{4 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (c_2 \left (e^{4 t}+1\right )-2 c_1 \left (e^{4 t}-1\right )\right ) \\ \end{align*}