76.27.6 problem 6

Internal problem ID [17852]
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 : 6
Date solved : Tuesday, January 28, 2025 at 11:04:03 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.064 (sec). Leaf size: 45

dsolve([diff(x__1(t),t)=-1*x__1(t)-4*x__2(t),diff(x__2(t),t)=1*x__1(t)-1*x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 55

DSolve[{D[x1[t],t]==-1*x1[t]-4*x2[t],D[x2[t],t]==1*x1[t]-1*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

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