9.6.10 problem problem 10

Internal problem ID [1017]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 10
Date solved : Saturday, March 29, 2025 at 10:36:57 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-13 x_{1} \left (t \right )+40 x_{2} \left (t \right )-48 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-8 x_{1} \left (t \right )+23 x_{2} \left (t \right )-24 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=3 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.133 (sec). Leaf size: 51

ode:=[diff(x__1(t),t) = -13*x__1(t)+40*x__2(t)-48*x__3(t), diff(x__2(t),t) = -8*x__1(t)+23*x__2(t)-24*x__3(t), diff(x__3(t),t) = 3*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{7 t}+c_2 \,{\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{7 t}}{2}+\frac {2 c_2 \,{\mathrm e}^{3 t}}{5}+\frac {6 c_3 \,{\mathrm e}^{3 t}}{5} \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 98

ode={D[ x1[t],t]==-13*x1[t]+40*x2[t]-48*x3[t],D[ x2[t],t]==-8*x1[t]+23*x2[t]-24*x3[t],D[ x3[t],t]==0*x1[t]+0*x2[t]+3*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.130 (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(13*x__1(t) - 40*x__2(t) + 48*x__3(t) + Derivative(x__1(t), t),0),Eq(8*x__1(t) - 23*x__2(t) + 24*x__3(t) + Derivative(x__2(t), t),0),Eq(-3*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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