76.27.10 problem 10

Internal problem ID [17856]
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 : 10
Date solved : Tuesday, January 28, 2025 at 11:04:07 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 83

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

\begin{align*} \text {x1}(t)\to \frac {1}{5} e^{-3 t/2} \left (c_1 \left (4 e^{5 t/2}+1\right )+c_2 \left (e^{5 t/2}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{5} e^{-3 t/2} \left (4 c_1 \left (e^{5 t/2}-1\right )+c_2 \left (e^{5 t/2}+4\right )\right ) \\ \end{align*}