76.27.9 problem 9

Internal problem ID [17855]
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 : 9
Date solved : Tuesday, January 28, 2025 at 11:04:06 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 30

dsolve([diff(x__1(t),t)=2*x__1(t)-1*x__2(t),diff(x__2(t),t)=3*x__1(t)-2*x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 73

DSolve[{D[x1[t],t]==2*x1[t]-1*x2[t],D[x2[t],t]==3*x1[t]-2*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-t} \left (c_1 \left (3 e^{2 t}-1\right )-c_2 \left (e^{2 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (3 c_1 \left (e^{2 t}-1\right )-c_2 \left (e^{2 t}-3\right )\right ) \\ \end{align*}