76.27.19 problem 19

Internal problem ID [17786]
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 : 19
Date solved : Thursday, March 13, 2025 at 10:49:16 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.069 (sec). Leaf size: 43

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 51

ode={D[x1[t],t]==-1*x1[t]-5*x2[t],D[x2[t],t]==1*x1[t]+3*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.094 (sec). Leaf size: 46

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) + Derivative(x__1(t), t),0),Eq(-x__1(t) - 3*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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