76.27.7 problem 7

Internal problem ID [17853]
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 : 7
Date solved : Tuesday, January 28, 2025 at 11:04:04 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 )&=3 x_{1} \left (t \right )+x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

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

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

\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{4 t} \\ x_{2} \left (t \right ) &= 3 c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{4 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 72

DSolve[{D[x1[t],t]==5*x1[t]-1*x2[t],D[x2[t],t]==13*x1[t]+1*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{3 t} (3 c_1 \cos (3 t)+(2 c_1-c_2) \sin (3 t)) \\ \text {x2}(t)\to \frac {1}{3} e^{3 t} (3 c_2 \cos (3 t)+(13 c_1-2 c_2) \sin (3 t)) \\ \end{align*}