9.6.8 problem problem 8

Internal problem ID [1015]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 8
Date solved : Saturday, March 29, 2025 at 10:36:54 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=25 x_{1} \left (t \right )+12 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-18 x_{1} \left (t \right )-5 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=6 x_{1} \left (t \right )+6 x_{2} \left (t \right )+13 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.177 (sec). Leaf size: 59

ode:=[diff(x__1(t),t) = 25*x__1(t)+12*x__2(t), diff(x__2(t),t) = -18*x__1(t)-5*x__2(t), diff(x__3(t),t) = 6*x__1(t)+6*x__2(t)+13*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{13 t}+c_3 \,{\mathrm e}^{7 t} \\ x_{2} \left (t \right ) &= -c_2 \,{\mathrm e}^{13 t}-\frac {3 c_3 \,{\mathrm e}^{7 t}}{2} \\ x_{3} \left (t \right ) &= \frac {c_2 \,{\mathrm e}^{13 t}}{2}+\frac {c_3 \,{\mathrm e}^{7 t}}{2}+{\mathrm e}^{13 t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 107

ode={D[ x1[t],t]==25*x1[t]+12*x2[t]+0*x3[t],D[ x2[t],t]==-18*x1[t]-5*x2[t]+0*x3[t],D[ x3[t],t]==6*x1[t]+6*x2[t]+13*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.126 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-25*x__1(t) - 12*x__2(t) + Derivative(x__1(t), t),0),Eq(18*x__1(t) + 5*x__2(t) + Derivative(x__2(t), t),0),Eq(-6*x__1(t) - 6*x__2(t) - 13*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = 2 C_{1} e^{7 t} - C_{2} e^{13 t}, \ x^{2}{\left (t \right )} = - 3 C_{1} e^{7 t} + C_{2} e^{13 t}, \ x^{3}{\left (t \right )} = C_{1} e^{7 t} + C_{3} e^{13 t}\right ] \]