14.22.12 problem 12

Internal problem ID [2739]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.8, Systems of differential equations. The eigenva1ue-eigenvector method. Page 339
Problem number : 12
Date solved : Tuesday, March 04, 2025 at 02:40:38 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )+x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=4 x_{1} \left (t \right )+x_{2} \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 ) = 4\\ x_{3} \left (0\right ) = -7 \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 57

ode:=[diff(x__1(t),t) = 3*x__1(t)+x__2(t)-2*x__3(t), diff(x__2(t),t) = -x__1(t)+2*x__2(t)+x__3(t), diff(x__3(t),t) = 4*x__1(t)+x__2(t)-3*x__3(t)]; 
ic:=x__1(0) = 1x__2(0) = 4x__3(0) = -7; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {4 \,{\mathrm e}^{2 t}}{3}+9 \,{\mathrm e}^{t}-\frac {28 \,{\mathrm e}^{-t}}{3} \\ x_{2} \left (t \right ) &= \frac {4 \,{\mathrm e}^{2 t}}{3}+\frac {8 \,{\mathrm e}^{-t}}{3} \\ x_{3} \left (t \right ) &= \frac {4 \,{\mathrm e}^{2 t}}{3}+9 \,{\mathrm e}^{t}-\frac {52 \,{\mathrm e}^{-t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 75

ode={D[ x1[t],t]==3*x1[t]+1*x2[t]-2*x3[t],D[ x2[t],t]==-1*x1[t]+2*x2[t]+1*x3[t],D[ x3[t],t]==4*x1[t]+1*x2[t]-3*x3[t]}; 
ic={x1[0]==1,x2[0]==4,x3[0]==-7}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to -\frac {28 e^{-t}}{3}+9 e^t+\frac {4 e^{2 t}}{3} \\ \text {x2}(t)\to \frac {4}{3} e^{-t} \left (e^{3 t}+2\right ) \\ \text {x3}(t)\to -\frac {52 e^{-t}}{3}+9 e^t+\frac {4 e^{2 t}}{3} \\ \end{align*}

✓ Sympy. Time used: 0.129 (sec). Leaf size: 58

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) - x__2(t) + 2*x__3(t) + Derivative(x__1(t), t),0),Eq(x__1(t) - 2*x__2(t) - x__3(t) + Derivative(x__2(t), t),0),Eq(-4*x__1(t) - x__2(t) + 3*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {7 C_{1} e^{- t}}{13} + C_{2} e^{t} + C_{3} e^{2 t}, \ x^{2}{\left (t \right )} = - \frac {2 C_{1} e^{- t}}{13} + C_{3} e^{2 t}, \ x^{3}{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{t} + C_{3} e^{2 t}\right ] \]