76.27.7 problem 7

Internal problem ID [17782]
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 : 7
Date solved : Monday, March 31, 2025 at 04:27:35 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.111 (sec). Leaf size: 34

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

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{4 t}+c_2 \,{\mathrm e}^{2 t} \\ x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{4 t}+3 c_2 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 72

ode={D[x1[t],t]==5*x1[t]-1*x2[t],D[x2[t],t]==13*x1[t]+1*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{3 t} (3 c_1 \cos (3 t)+(2 c_1-c_2) \sin (3 t)) \\ \text {x2}(t)\to \frac {1}{3} e^{3 t} (3 c_2 \cos (3 t)+(13 c_1-2 c_2) \sin (3 t)) \\ \end{align*}

✓ Sympy. Time used: 0.082 (sec). Leaf size: 32

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

\[ \left [ x^{1}{\left (t \right )} = \frac {C_{1} e^{2 t}}{3} + C_{2} e^{4 t}, \ x^{2}{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{4 t}\right ] \]