76.25.6 problem 6

Internal problem ID [17740]
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 : 6
Date solved : Thursday, March 13, 2025 at 10:48:13 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 66

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

\begin{align*} x_{1} \left (t \right ) &= 2 c_{2} {\mathrm e}^{-t}+2 c_{3} {\mathrm e}^{8 t}+{\mathrm e}^{-t} c_{1} \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{8 t} \\ x_{3} \left (t \right ) &= -\frac {5 c_{2} {\mathrm e}^{-t}}{2}+2 c_{3} {\mathrm e}^{8 t}-{\mathrm e}^{-t} c_{1} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 135

ode={D[x1[t],t]==3*x1[t]+2*x2[t]+4*x3[t],D[x2[t],t]==2*x1[t]+0*x2[t]+2*x3[t],D[x3[t],t]==4*x1[t]+2*x2[t]+3*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{9} e^{-t} \left (c_1 \left (4 e^{9 t}+5\right )+2 (c_2+2 c_3) \left (e^{9 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-t} \left (2 c_1 \left (e^{9 t}-1\right )+c_2 \left (e^{9 t}+8\right )+2 c_3 \left (e^{9 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{9} e^{-t} \left (4 c_1 \left (e^{9 t}-1\right )+2 c_2 \left (e^{9 t}-1\right )+c_3 \left (4 e^{9 t}+5\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.123 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-3*x__1(t) - 2*x__2(t) - 4*x__3(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - 2*x__3(t) + Derivative(x__2(t), t),0),Eq(-4*x__1(t) - 2*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)
 

\[ \left [ x^{1}{\left (t \right )} = C_{3} e^{8 t} - \left (C_{1} + \frac {C_{2}}{2}\right ) e^{- t}, \ x^{2}{\left (t \right )} = C_{2} e^{- t} + \frac {C_{3} e^{8 t}}{2}, \ x^{3}{\left (t \right )} = C_{1} e^{- t} + C_{3} e^{8 t}\right ] \]