76.27.8 problem 8

Internal problem ID [17854]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.5 (Fundamental Matrices and the Exponential of a Matrix). Problems at page 430
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 11:04:05 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=5 x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.063 (sec). Leaf size: 46

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

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 56

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

\begin{align*} \text {x1}(t)\to e^{-t} (c_1 \cos (t)+(2 c_1-c_2) \sin (t)) \\ \text {x2}(t)\to e^{-t} (c_2 \cos (t)+(5 c_1-2 c_2) \sin (t)) \\ \end{align*}