14.24.3 problem 1

Internal problem ID [2750]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number : 1
Date solved : Sunday, March 30, 2025 at 12:16:24 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.175 (sec). Leaf size: 53

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_3 t +c_2 \right ) \\ x_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{-t} t +{\mathrm e}^{-2 t} c_1 \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{-t}+{\mathrm e}^{-2 t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 99

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

\begin{align*} \text {x1}(t)\to e^{-t} (c_1 (t+1)+(c_3-c_2) t) \\ \text {x2}(t)\to e^{-2 t} \left (c_1 \left (e^t (t+1)-1\right )-c_2 e^t t+c_3 e^t t+c_2\right ) \\ \text {x3}(t)\to e^{-2 t} \left (c_1 \left (e^t-1\right )-c_2 e^t+c_3 e^t+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.124 (sec). Leaf size: 48

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

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