14.24.7 problem 5

Internal problem ID [2754]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number : 5
Date solved : Tuesday, March 04, 2025 at 02:40:54 PM
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 )+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 )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 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 ) = 0\\ x_{3} \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 15

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \\ x_{2} \left (t \right ) &= 0 \\ x_{3} \left (t \right ) &= {\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 18

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

\begin{align*} \text {x1}(t)\to e^t \\ \text {x2}(t)\to 0 \\ \text {x3}(t)\to e^t \\ \end{align*}

✓ Sympy. Time used: 0.164 (sec). Leaf size: 82

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) - 2*x__3(t) + Derivative(x__1(t), t),0),Eq(x__1(t) - x__2(t) - x__3(t) + Derivative(x__2(t), t),0),Eq(2*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)
 

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