76.27.1 problem 1

Internal problem ID [17847]
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 : 1
Date solved : Tuesday, January 28, 2025 at 11:04:00 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 73

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

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (4 e^{3 t}-1\right )-2 c_2 \left (e^{3 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-t} \left (2 c_1 \left (e^{3 t}-1\right )-c_2 \left (e^{3 t}-4\right )\right ) \\ \end{align*}