14.23.7 problem 7

Internal problem ID [2746]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number : 7
Date solved : Tuesday, March 04, 2025 at 02:40:45 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_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.054 (sec). Leaf size: 94

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

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

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 109

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

\begin{align*} \text {x1}(t)\to -e^{-2 t} \left (\sqrt {2} e^t \sin \left (\sqrt {2} t\right )+2 e^t \cos \left (\sqrt {2} t\right )-2\right ) \\ \text {x2}(t)\to e^{-2 t} \left (-\sqrt {2} e^t \sin \left (\sqrt {2} t\right )+e^t \cos \left (\sqrt {2} t\right )-2\right ) \\ \text {x3}(t)\to e^{-2 t} \left (1-3 e^t \cos \left (\sqrt {2} t\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.291 (sec). Leaf size: 155

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

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