14.22.10 problem 10

Internal problem ID [2737]
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 : 10
Date solved : Sunday, March 30, 2025 at 12:16:03 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 )&=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 )&=x_{1} \left (t \right )+10 x_{2} \left (t \right )+2 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 ) = 13 \end{align*}

✓ Maple. Time used: 0.139 (sec). Leaf size: 36

ode:=[diff(x__1(t),t) = x__1(t)-x__2(t), diff(x__2(t),t) = x__1(t)+2*x__2(t)+x__3(t), diff(x__3(t),t) = x__1(t)+10*x__2(t)+2*x__3(t)]; 
ic:=x__1(0) = -1x__2(0) = -4x__3(0) = 13; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t}-2 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= -4 \,{\mathrm e}^{-t} \\ x_{3} \left (t \right ) &= -{\mathrm e}^{t}+14 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 42

ode={D[ x1[t],t]==1*x1[t]-1*x2[t]-0*x3[t],D[ x2[t],t]==1*x1[t]+2*x2[t]+1*x3[t],D[ x3[t],t]==1*x1[t]+10*x2[t]+2*x3[t]}; 
ic={x1[0]==-1,x2[0]==-4,x3[0]==13}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^t-2 e^{-t} \\ \text {x2}(t)\to -4 e^{-t} \\ \text {x3}(t)\to 14 e^{-t}-e^t \\ \end{align*}

✓ Sympy. Time used: 0.147 (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") 
ode=[Eq(-x__1(t) + x__2(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(-x__1(t) - 10*x__2(t) - 2*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {C_{1} e^{- t}}{7} - C_{2} e^{t} - \frac {C_{3} e^{5 t}}{13}, \ x^{2}{\left (t \right )} = - \frac {2 C_{1} e^{- t}}{7} + \frac {4 C_{3} e^{5 t}}{13}, \ x^{3}{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{t} + C_{3} e^{5 t}\right ] \]