14.24.5 problem 3

Internal problem ID [2752]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number : 3
Date solved : Sunday, March 30, 2025 at 12:16:26 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_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.162 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 74

ode={D[ x1[t],t]==-1*x1[t]-1*x2[t]+0*x3[t],D[ x2[t],t]==0*x1[t]-1*x2[t]+0*x3[t],D[ x3[t],t]==0*x1[t]-0*x2[t]-2*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-c_2 t) \\ \text {x2}(t)\to c_2 e^{-t} \\ \text {x3}(t)\to c_3 e^{-2 t} \\ \text {x1}(t)\to e^{-t} (c_1-c_2 t) \\ \text {x2}(t)\to c_2 e^{-t} \\ \text {x3}(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.088 (sec). Leaf size: 31

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__2(t) + Derivative(x__2(t), t),0),Eq(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)
 

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