9.4.20 problem problem 20

Internal problem ID [984]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 20
Date solved : Tuesday, March 04, 2025 at 12:06:50 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 63

ode:=[diff(x__1(t),t) = 5*x__1(t)+x__2(t)+3*x__3(t), diff(x__2(t),t) = x__1(t)+7*x__2(t)+x__3(t), diff(x__3(t),t) = 3*x__1(t)+x__2(t)+5*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{9 t}+c_2 \,{\mathrm e}^{2 t}+c_3 \,{\mathrm e}^{6 t} \\ x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{9 t}-2 c_3 \,{\mathrm e}^{6 t} \\ x_{3} \left (t \right ) &= c_1 \,{\mathrm e}^{9 t}-c_2 \,{\mathrm e}^{2 t}+c_3 \,{\mathrm e}^{6 t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 163

ode={D[ x1[t],t]==5*x1[t]+1*x2[t]+3*x3[t],D[ x2[t],t]==1*x1[t]+7*x2[t]+1*x3[t],D[ x3[t],t]==3*x1[t]+1*x2[t]+5*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^{4 t}+2 e^{7 t}+3\right )+(c_3-2 c_2) e^{4 t}+2 (c_2+c_3) e^{7 t}-3 c_3\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^{2 t} \left (c_1 \left (e^{4 t}+2 e^{7 t}-3\right )+(c_3-2 c_2) e^{4 t}+2 (c_2+c_3) e^{7 t}+3 c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.146 (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(-5*x__1(t) - x__2(t) - 3*x__3(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) - 7*x__2(t) - x__3(t) + Derivative(x__2(t), t),0),Eq(-3*x__1(t) - x__2(t) - 5*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - C_{1} e^{2 t} + C_{2} e^{6 t} + C_{3} e^{9 t}, \ x^{2}{\left (t \right )} = - 2 C_{2} e^{6 t} + C_{3} e^{9 t}, \ x^{3}{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{6 t} + C_{3} e^{9 t}\right ] \]