76.25.4 problem 4

Internal problem ID [17738]
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 6.3 (Homogeneous Linear Systems with Constant Coefficients). Problems at page 408
Problem number : 4
Date solved : Thursday, March 13, 2025 at 10:48:11 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-2 x_{1} \left (t \right )+2 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 )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{1} \left (t \right )+4 x_{2} \left (t \right )-3 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.134 (sec). Leaf size: 53

ode:=[diff(x__1(t),t) = -2*x__1(t)+2*x__2(t)-x__3(t), diff(x__2(t),t) = -2*x__1(t)+3*x__2(t)-2*x__3(t), diff(x__3(t),t) = -2*x__1(t)+4*x__2(t)-3*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 112

ode={D[x1[t],t]==-2*x1[t]+2*x2[t]-1*x3[t],D[x2[t],t]==-2*x1[t]+3*x2[t]-2*x3[t],D[x3[t],t]==-2*x1[t]+4*x2[t]-3*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.111 (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(2*x__1(t) - 2*x__2(t) + x__3(t) + Derivative(x__1(t), t),0),Eq(2*x__1(t) - 3*x__2(t) + 2*x__3(t) + Derivative(x__2(t), t),0),Eq(2*x__1(t) - 4*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)
 

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