9.6.9 problem problem 9

Internal problem ID [1016]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 9
Date solved : Saturday, March 29, 2025 at 10:36:56 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-19 x_{1} \left (t \right )+12 x_{2} \left (t \right )+84 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 )&=-8 x_{1} \left (t \right )+4 x_{2} \left (t \right )+33 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.137 (sec). Leaf size: 51

ode:=[diff(x__1(t),t) = -19*x__1(t)+12*x__2(t)+84*x__3(t), diff(x__2(t),t) = 5*x__2(t), diff(x__3(t),t) = -8*x__1(t)+4*x__2(t)+33*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{9 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}^{9 t}}{3}+\frac {2 c_2 \,{\mathrm e}^{5 t}}{7}-\frac {c_3 \,{\mathrm e}^{5 t}}{7} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 94

ode={D[ x1[t],t]==-19*x1[t]+12*x2[t]+84*x3[t],D[ x2[t],t]==0*x1[t]+5*x2[t]+0*x3[t],D[ x3[t],t]==-8*x1[t]+4*x2[t]+33*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{5 t} \left (c_1 \left (7-6 e^{4 t}\right )+3 (c_2+7 c_3) \left (e^{4 t}-1\right )\right ) \\ \text {x2}(t)\to c_2 e^{5 t} \\ \text {x3}(t)\to e^{5 t} \left (-2 c_1 \left (e^{4 t}-1\right )+c_2 \left (e^{4 t}-1\right )+c_3 \left (7 e^{4 t}-6\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.125 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(19*x__1(t) - 12*x__2(t) - 84*x__3(t) + Derivative(x__1(t), t),0),Eq(-5*x__2(t) + Derivative(x__2(t), t),0),Eq(8*x__1(t) - 4*x__2(t) - 33*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = 3 C_{3} e^{9 t} + \left (\frac {C_{1}}{2} + \frac {7 C_{2}}{2}\right ) e^{5 t}, \ x^{2}{\left (t \right )} = C_{1} e^{5 t}, \ x^{3}{\left (t \right )} = C_{2} e^{5 t} + C_{3} e^{9 t}\right ] \]