76.27.4 problem 4

Internal problem ID [17850]
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 : 4
Date solved : Tuesday, January 28, 2025 at 11:04:02 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 )}{4}\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-\frac {x_{2} \left (t \right )}{2} \end{align*}

✓ Solution by Maple

Time used: 0.095 (sec). Leaf size: 23

dsolve([diff(x__1(t),t)=1/2*x__1(t)-1/4*x__2(t),diff(x__2(t),t)=1*x__1(t)-1/2*x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 39

DSolve[{D[x1[t],t]==1/2*x1[t]-1/4*x2[t],D[x2[t],t]==1*x1[t]-1/2*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

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