14.23.5 problem 5

Internal problem ID [2744]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number : 5
Date solved : Monday, January 27, 2025 at 06:12:37 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 )&=5 x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = 2 \end{align*}

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 26

dsolve([diff(x__1(t),t) = x__1(t)-x__2(t), diff(x__2(t),t) = 5*x__1(t)-3*x__2(t), x__1(0) = 1, x__2(0) = 2], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 29

DSolve[{D[ x1[t],t]==1*x1[t]-1*x2[t],D[ x2[t],t]==5*x1[t]-3*x2[t]},{x1[0]==1,x2[0]==2},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-t} \cos (t) \\ \text {x2}(t)\to e^{-t} (\sin (t)+2 \cos (t)) \\ \end{align*}