76.27.12 problem 12

Internal problem ID [17787]
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 : 12
Date solved : Monday, March 31, 2025 at 04:27:42 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )+\frac {5 x_{2} \left (t \right )}{2}\\ \frac {d}{d t}x_{2} \left (t \right )&=-\frac {5 x_{1} \left (t \right )}{2}+2 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.105 (sec). Leaf size: 34

ode:=[diff(x__1(t),t) = -3*x__1(t)+5/2*x__2(t), diff(x__2(t),t) = -5/2*x__1(t)+2*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (c_2 t +c_1 \right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-\frac {t}{2}} \left (5 c_2 t +5 c_1 +2 c_2 \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 59

ode={D[x1[t],t]==-3*x1[t]+5/2*x2[t],D[x2[t],t]==-5/2*x1[t]+2*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.112 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(3*x__1(t) - 5*x__2(t)/2 + Derivative(x__1(t), t),0),Eq(5*x__1(t)/2 - 2*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {5 C_{2} t e^{- \frac {t}{2}}}{2} - \left (\frac {5 C_{1}}{2} - C_{2}\right ) e^{- \frac {t}{2}}, \ x^{2}{\left (t \right )} = - \frac {5 C_{1} e^{- \frac {t}{2}}}{2} - \frac {5 C_{2} t e^{- \frac {t}{2}}}{2}\right ] \]