76.27.14 problem 14

Internal problem ID [17781]
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 6.5 (Fundamental Matrices and the Exponential of a Matrix). Problems at page 430
Problem number : 14
Date solved : Thursday, March 13, 2025 at 10:49:11 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 )+4 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{1} \left (t \right )+2 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 )-x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 66

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

\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+{\mathrm e}^{-2 t} c_{2} +c_{3} {\mathrm e}^{t} \\ x_{2} \left (t \right ) &= 2 c_{1} {\mathrm e}^{3 t}-{\mathrm e}^{-2 t} c_{2} -4 c_{3} {\mathrm e}^{t} \\ x_{3} \left (t \right ) &= c_{1} {\mathrm e}^{3 t}-{\mathrm e}^{-2 t} c_{2} -c_{3} {\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 182

ode={D[x1[t],t]==1*x1[t]-1*x2[t]+4*x3[t],D[x2[t],t]==3*x1[t]+2*x2[t]-1*x3[t],D[x3[t],t]==2*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 \frac {1}{6} e^{-2 t} \left (c_1 \left (e^{3 t}+3 e^{5 t}+2\right )-2 c_2 \left (e^{3 t}-1\right )+3 c_3 \left (e^{3 t}+e^{5 t}-2\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-2 t} \left (c_1 \left (-2 e^{3 t}+3 e^{5 t}-1\right )+c_2 \left (4 e^{3 t}-1\right )+3 c_3 \left (-2 e^{3 t}+e^{5 t}+1\right )\right ) \\ \text {x3}(t)\to \frac {1}{6} e^{-2 t} \left (c_1 \left (-e^{3 t}+3 e^{5 t}-2\right )+2 c_2 \left (e^{3 t}-1\right )+3 c_3 \left (-e^{3 t}+e^{5 t}+2\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.152 (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) - 4*x__3(t) + Derivative(x__1(t), t),0),Eq(-3*x__1(t) - 2*x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(-2*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)
 

\[ \left [ x^{1}{\left (t \right )} = - C_{1} e^{- 2 t} - C_{2} e^{t} + C_{3} e^{3 t}, \ x^{2}{\left (t \right )} = C_{1} e^{- 2 t} + 4 C_{2} e^{t} + 2 C_{3} e^{3 t}, \ x^{3}{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{t} + C_{3} e^{3 t}\right ] \]