14.23.8 problem 8

Internal problem ID [2747]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number : 8
Date solved : Sunday, March 30, 2025 at 12:16:19 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = 1\\ x_{3} \left (0\right ) = 1\\ x_{4} \left (0\right ) = 0 \end{align*}

✓ Maple. Time used: 0.178 (sec). Leaf size: 41

ode:=[diff(x__1(t),t) = 2*x__2(t), diff(x__2(t),t) = -2*x__1(t), diff(x__3(t),t) = -3*x__4(t), diff(x__4(t),t) = 3*x__3(t)]; 
ic:=x__1(0) = 1x__2(0) = 1x__3(0) = 1x__4(0) = 0; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 42

ode={D[ x1[t],t]==-0*x1[t]+2*x2[t]+0*x3[t]+0*x4[t],D[ x2[t],t]==-2*x1[t]-0*x2[t]-0*x3[t]+0*x4[t],D[ x3[t],t]==0*x1[t]-0*x2[t]-0*x3[t]-3*x4[t],D[ x4[t],t]==0*x1[t]-0*x2[t]+3*x3[t]-0*x4[t]}; 
ic={x1[0]==1,x2[0]==1,x3[0]==1,x4[0]==0}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \sin (2 t)+\cos (2 t) \\ \text {x2}(t)\to \cos (2 t)-\sin (2 t) \\ \text {x3}(t)\to \cos (3 t) \\ \text {x4}(t)\to \sin (3 t) \\ \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
x__4 = Function("x__4") 
ode=[Eq(-2*x__2(t) + Derivative(x__1(t), t),0),Eq(2*x__1(t) + Derivative(x__2(t), t),0),Eq(3*x__4(t) + Derivative(x__3(t), t),0),Eq(-3*x__3(t) + Derivative(x__4(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
 

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