76.23.1 problem 2

Internal problem ID [17721]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.1 (Definitions and examples). Problems at page 388
Problem number : 2
Date solved : Monday, March 31, 2025 at 04:25:59 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 )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{2} \left (t \right )+x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.113 (sec). Leaf size: 75

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

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

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 164

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

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (2 e^{3 t}+1\right )+(c_2+c_3) \left (e^{3 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-t} \left (c_1 \left (e^{3 t} (6 t+4)-4\right )+c_2 \left (e^{3 t} (3 t+5)+4\right )+c_3 \left (e^{3 t} (3 t-4)+4\right )\right ) \\ \text {x3}(t)\to \frac {1}{9} e^{-t} \left (c_1 \left (e^{3 t} (2-6 t)-2\right )+c_2 \left (2-e^{3 t} (3 t+2)\right )-c_3 \left (e^{3 t} (3 t-7)-2\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 65

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

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