14.23.6 problem 6

Internal problem ID [2745]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number : 6
Date solved : Tuesday, March 04, 2025 at 02:40:44 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = 5 \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 37

ode:=[diff(x__1(t),t) = 3*x__1(t)-2*x__2(t), diff(x__2(t),t) = 4*x__1(t)-x__2(t)]; 
ic:=x__1(0) = 1x__2(0) = 5; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (-4 \sin \left (2 t \right )+\cos \left (2 t \right )\right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} \left (-3 \sin \left (2 t \right )+5 \cos \left (2 t \right )\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 40

ode={D[ x1[t],t]==3*x1[t]-2*x2[t],D[ x2[t],t]==4*x1[t]-1*x2[t]}; 
ic={x1[0]==1,x2[0]==5}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^t (\cos (2 t)-4 \sin (2 t)) \\ \text {x2}(t)\to e^t (5 \cos (2 t)-3 \sin (2 t)) \\ \end{align*}

✓ Sympy. Time used: 0.107 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-3*x__1(t) + 2*x__2(t) + Derivative(x__1(t), t),0),Eq(-4*x__1(t) + x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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