9.6.7 problem problem 7

Internal problem ID [1014]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 7
Date solved : Saturday, March 29, 2025 at 10:36:53 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-7 x_{1} \left (t \right )+9 x_{2} \left (t \right )+7 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.178 (sec). Leaf size: 42

ode:=[diff(x__1(t),t) = 2*x__1(t), diff(x__2(t),t) = -7*x__1(t)+9*x__2(t)+7*x__3(t), diff(x__3(t),t) = 2*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 60

ode={D[ x1[t],t]==2*x1[t]+0*x2[t]+0*x3[t],D[ x2[t],t]==-7*x1[t]+9*x2[t]+7*x3[t],D[ x3[t],t]==0*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 c_1 e^{2 t} \\ \text {x2}(t)\to e^{2 t} \left (-\left (c_1 \left (e^{7 t}-1\right )\right )+(c_2+c_3) e^{7 t}-c_3\right ) \\ \text {x3}(t)\to c_3 e^{2 t} \\ \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-2*x__1(t) + Derivative(x__1(t), t),0),Eq(7*x__1(t) - 9*x__2(t) - 7*x__3(t) + Derivative(x__2(t), t),0),Eq(-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)
 

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