14.23.5 problem 5

Internal problem ID [2744]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number : 5
Date solved : Sunday, March 30, 2025 at 12:16:14 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = 2 \end{align*}

✓ Maple. Time used: 0.157 (sec). Leaf size: 26

ode:=[diff(x__1(t),t) = x__1(t)-x__2(t), diff(x__2(t),t) = 5*x__1(t)-3*x__2(t)]; 
ic:=x__1(0) = 1x__2(0) = 2; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 29

ode={D[ x1[t],t]==1*x1[t]-1*x2[t],D[ x2[t],t]==5*x1[t]-3*x2[t]}; 
ic={x1[0]==1,x2[0]==2}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.107 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t) + x__2(t) + Derivative(x__1(t), t),0),Eq(-5*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)
 

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