76.27.2 problem 2

Internal problem ID [17848]
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 : 2
Date solved : Tuesday, January 28, 2025 at 11:04:01 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 )&=\frac {x_{1} \left (t \right )}{2}-3 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.075 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 70

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

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