76.27.12 problem 12

Internal problem ID [17858]
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 : 12
Date solved : Tuesday, January 28, 2025 at 11:04:08 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 59

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

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-t/2} (c_1 (2-5 t)+5 c_2 t) \\ \text {x2}(t)\to \frac {1}{2} e^{-t/2} (-5 c_1 t+5 c_2 t+2 c_2) \\ \end{align*}