76.27.18 problem 18

Internal problem ID [17864]
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 : 18
Date solved : Tuesday, January 28, 2025 at 11:04:13 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.055 (sec). Leaf size: 30

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{4 t} \left (c_{2} t +c_{1} \right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{4 t} \left (c_{2} t +c_{1} -c_{2} \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 44

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

\begin{align*} \text {x1}(t)\to e^{4 t} (c_1 (t+1)-c_2 t) \\ \text {x2}(t)\to e^{4 t} ((c_1-c_2) t+c_2) \\ \end{align*}