28.4.31 problem 7.31

Internal problem ID [4563]
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 : 7.31
Date solved : Monday, January 27, 2025 at 09:23:32 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 40

dsolve([diff(x__1(t),t)=x__1(t)-3*x__2(t),diff(x__2(t),t)=3*x__1(t)+x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 47

DSolve[{D[x1[t],t]==x1[t]-3*x2[t],D[x2[t],t]==3*x1[t]+x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^t (c_1 \cos (3 t)-c_2 \sin (3 t)) \\ \text {x2}(t)\to e^t (c_2 \cos (3 t)+c_1 \sin (3 t)) \\ \end{align*}