76.25.1 problem 1

Internal problem ID [17735]
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 : 1
Date solved : Thursday, March 13, 2025 at 10:48:08 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-4 x_{1} \left (t \right )+x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-5 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{2} \left (t \right )-4 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.239 (sec). Leaf size: 64

ode:=[diff(x__1(t),t) = -4*x__1(t)+x__2(t), diff(x__2(t),t) = x__1(t)-5*x__2(t)+x__3(t), diff(x__3(t),t) = x__2(t)-4*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} c_{2} +{\mathrm e}^{-4 t} c_{1} -\frac {c_{3} {\mathrm e}^{-6 t}}{2} \\ x_{2} \left (t \right ) &= {\mathrm e}^{-3 t} c_{2} +c_{3} {\mathrm e}^{-6 t} \\ x_{3} \left (t \right ) &= {\mathrm e}^{-3 t} c_{2} -{\mathrm e}^{-4 t} c_{1} -\frac {c_{3} {\mathrm e}^{-6 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 165

ode={D[x1[t],t]==-4*x1[t]+x2[t]+0*x3[t],D[x2[t],t]==1*x1[t]-5*x2[t]+1*x3[t],D[x3[t],t]==0*x1[t]+x2[t]-4*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^{-6 t} \left (c_3 \left (2 e^t+1\right ) \left (e^t-1\right )^2+c_1 \left (3 e^{2 t}+2 e^{3 t}+1\right )+2 c_2 \left (e^{3 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-6 t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}+2\right )+c_3 \left (e^{3 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{6} e^{-6 t} \left (c_1 \left (2 e^t+1\right ) \left (e^t-1\right )^2+2 c_2 \left (e^{3 t}-1\right )+c_3 \left (3 e^{2 t}+2 e^{3 t}+1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.141 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(4*x__1(t) - x__2(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + 5*x__2(t) - x__3(t) + Derivative(x__2(t), t),0),Eq(-x__2(t) + 4*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{- 6 t} - C_{2} e^{- 4 t} + C_{3} e^{- 3 t}, \ x^{2}{\left (t \right )} = - 2 C_{1} e^{- 6 t} + C_{3} e^{- 3 t}, \ x^{3}{\left (t \right )} = C_{1} e^{- 6 t} + C_{2} e^{- 4 t} + C_{3} e^{- 3 t}\right ] \]