76.27.5 problem 5

Internal problem ID [17851]
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 : 5
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 )-\frac {5 x_{2} \left (t \right )}{2}\\ \frac {d}{d t}x_{2} \left (t \right )&=\frac {x_{1} \left (t \right )}{2}-x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 49

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

\begin{align*} x_{1} \left (t \right ) &= c_{1} \sin \left (\frac {t}{2}\right )+c_{2} \cos \left (\frac {t}{2}\right ) \\ x_{2} \left (t \right ) &= -\frac {\cos \left (\frac {t}{2}\right ) c_{1}}{5}+\frac {\sin \left (\frac {t}{2}\right ) c_{2}}{5}+\frac {2 c_{1} \sin \left (\frac {t}{2}\right )}{5}+\frac {2 c_{2} \cos \left (\frac {t}{2}\right )}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 58

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

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