14.22.4 problem 4

Internal problem ID [2731]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.8, Systems of differential equations. The eigenva1ue-eigenvector method. Page 339
Problem number : 4
Date solved : Sunday, March 30, 2025 at 12:15:55 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=7 x_{1} \left (t \right )-x_{2} \left (t \right )+6 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-10 x_{1} \left (t \right )+4 x_{2} \left (t \right )-12 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: 0.109 (sec). Leaf size: 73

ode:=[diff(x__1(t),t) = 7*x__1(t)-x__2(t)+6*x__3(t), diff(x__2(t),t) = -10*x__1(t)+4*x__2(t)-12*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}+c_2 \,{\mathrm e}^{5 t}+c_3 \,{\mathrm e}^{2 t} \\ x_{2} \left (t \right ) &= -2 c_1 \,{\mathrm e}^{3 t}-2 c_2 \,{\mathrm e}^{5 t}-c_3 \,{\mathrm e}^{2 t} \\ x_{3} \left (t \right ) &= -c_1 \,{\mathrm e}^{3 t}-\frac {2 c_2 \,{\mathrm e}^{5 t}}{3}-c_3 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 153

ode={D[ x1[t],t]==7*x1[t]-1*x2[t]+6*x3[t],D[ x2[t],t]==-10*x1[t]+4*x2[t]-12*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 e^{2 t} \left (c_1 \left (-4 e^t+3 e^{3 t}+2\right )-c_2 \left (e^t-1\right )+3 c_3 e^t \left (e^{2 t}-1\right )\right ) \\ \text {x2}(t)\to -e^{2 t} \left (c_1 \left (-8 e^t+6 e^{3 t}+2\right )+c_2 \left (1-2 e^t\right )+6 c_3 e^t \left (e^{2 t}-1\right )\right ) \\ \text {x3}(t)\to e^{2 t} \left (-2 c_1 \left (-2 e^t+e^{3 t}+1\right )+c_2 \left (e^t-1\right )+c_3 e^t \left (3-2 e^{2 t}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.156 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-7*x__1(t) + x__2(t) - 6*x__3(t) + Derivative(x__1(t), t),0),Eq(10*x__1(t) - 4*x__2(t) + 12*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)
 

\[ \left [ x^{1}{\left (t \right )} = - C_{1} e^{2 t} - C_{2} e^{3 t} - \frac {3 C_{3} e^{5 t}}{2}, \ x^{2}{\left (t \right )} = C_{1} e^{2 t} + 2 C_{2} e^{3 t} + 3 C_{3} e^{5 t}, \ x^{3}{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{3 t} + C_{3} e^{5 t}\right ] \]