7.25.17 problem 17

Internal problem ID [637]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.4 (The eigenvalue method for homogeneous systems). Problems at page 378
Problem number : 17
Date solved : Saturday, March 29, 2025 at 05:01:16 PM
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 )+4 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 )&=4 x_{1} \left (t \right )+x_{2} \left (t \right )+4 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.120 (sec). Leaf size: 54

ode:=[diff(x__1(t),t) = 4*x__1(t)+x__2(t)+4*x__3(t), diff(x__2(t),t) = x__1(t)+7*x__2(t)+x__3(t), diff(x__3(t),t) = 4*x__1(t)+x__2(t)+4*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 158

ode={D[x1[t],t]==4*x1[t]+x2[t]+4*x3[t],D[x2[t],t]==x1[t]+7*x2[t]+x3[t],D[x3[t],t]==4*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} \left (c_1 \left (e^{6 t}+2 e^{9 t}+3\right )+\left (e^{3 t}-1\right ) \left (3 c_3 e^{3 t}+2 (c_2+c_3) e^{6 t}+3 c_3\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} \left (c_1 \left (e^{6 t}+2 e^{9 t}-3\right )+(c_3-2 c_2) e^{6 t}+2 (c_2+c_3) e^{9 t}+3 c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.277 (sec). Leaf size: 51

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) - 4*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(-4*x__1(t) - 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)
 

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