76.27.4 problem 4

Internal problem ID [17779]
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 : Monday, March 31, 2025 at 04:27:31 PM
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*}

✓ Maple. Time used: 0.082 (sec). Leaf size: 23

ode:=[diff(x__1(t),t) = 1/2*x__1(t)-1/4*x__2(t), diff(x__2(t),t) = x__1(t)-1/2*x__2(t)]; 
dsolve(ode);
 

\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*}

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

ode={D[x1[t],t]==1/2*x1[t]-1/4*x2[t],D[x2[t],t]==1*x1[t]-1/2*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.068 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t)/2 + x__2(t)/4 + Derivative(x__1(t), t),0),Eq(-x__1(t) + x__2(t)/2 + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {C_{1}}{2} + \frac {C_{2} t}{2} + C_{2}, \ x^{2}{\left (t \right )} = C_{1} + C_{2} t\right ] \]