14.22.5 problem 5

Internal problem ID [2732]
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 : 5
Date solved : Sunday, March 30, 2025 at 12:15:56 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-7 x_{1} \left (t \right )+6 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=5 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=6 x_{1} \left (t \right )+2 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.137 (sec). Leaf size: 44

ode:=[diff(x__1(t),t) = -7*x__1(t)+6*x__3(t), diff(x__2(t),t) = 5*x__2(t), diff(x__3(t),t) = 6*x__1(t)+2*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-10 t}+c_2 \,{\mathrm e}^{5 t} \\ x_{2} \left (t \right ) &= c_3 \,{\mathrm e}^{5 t} \\ x_{3} \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{-10 t}}{2}+2 c_2 \,{\mathrm e}^{5 t} \\ \end{align*}

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 158

ode={D[ x1[t],t]==-7*x1[t]+0*x2[t]+6*x3[t],D[ x2[t],t]==0*x1[t]+5*x2[t]+0*x3[t],D[ x3[t],t]==6*x1[t]+0*x2[t]+2*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{5} e^{-10 t} \left (c_1 \left (e^{15 t}+4\right )+2 c_2 \left (e^{15 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{5} e^{-10 t} \left (2 c_1 \left (e^{15 t}-1\right )+c_2 \left (4 e^{15 t}+1\right )\right ) \\ \text {x2}(t)\to c_3 e^{5 t} \\ \text {x1}(t)\to \frac {1}{5} e^{-10 t} \left (c_1 \left (e^{15 t}+4\right )+2 c_2 \left (e^{15 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{5} e^{-10 t} \left (2 c_1 \left (e^{15 t}-1\right )+c_2 \left (4 e^{15 t}+1\right )\right ) \\ \text {x2}(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.126 (sec). Leaf size: 42

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) - 6*x__3(t) + Derivative(x__1(t), t),0),Eq(-5*x__2(t) + Derivative(x__2(t), t),0),Eq(-6*x__1(t) - 2*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - 2 C_{1} e^{- 10 t} + \frac {C_{2} e^{5 t}}{2}, \ x^{2}{\left (t \right )} = C_{3} e^{5 t}, \ x^{3}{\left (t \right )} = C_{1} e^{- 10 t} + C_{2} e^{5 t}\right ] \]