76.27.5 problem 5

Internal problem ID [17772]
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 : 5
Date solved : Thursday, March 13, 2025 at 10:49:02 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-\frac {5 x_{2} \left (t \right )}{2}\\ \frac {d}{d t}x_{2} \left (t \right )&=\frac {x_{1} \left (t \right )}{2}-x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.060 (sec). Leaf size: 49

ode:=[diff(x__1(t),t) = x__1(t)-5/2*x__2(t), diff(x__2(t),t) = 1/2*x__1(t)-x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_{1} \sin \left (\frac {t}{2}\right )+c_{2} \cos \left (\frac {t}{2}\right ) \\ x_{2} \left (t \right ) &= -\frac {\cos \left (\frac {t}{2}\right ) c_{1}}{5}+\frac {\sin \left (\frac {t}{2}\right ) c_{2}}{5}+\frac {2 c_{1} \sin \left (\frac {t}{2}\right )}{5}+\frac {2 c_{2} \cos \left (\frac {t}{2}\right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 58

ode={D[x1[t],t]==1*x1[t]-5/2*x2[t],D[x2[t],t]==1/2*x1[t]-1*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to c_1 \cos \left (\frac {t}{2}\right )+(2 c_1-5 c_2) \sin \left (\frac {t}{2}\right ) \\ \text {x2}(t)\to c_2 \cos \left (\frac {t}{2}\right )+(c_1-2 c_2) \sin \left (\frac {t}{2}\right ) \\ \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t) + 5*x__2(t)/2 + Derivative(x__1(t), t),0),Eq(-x__1(t)/2 + x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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