9.6.23 problem problem 23

Internal problem ID [1030]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 23
Date solved : Tuesday, March 04, 2025 at 12:07:43 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=39 x_{1} \left (t \right )+8 x_{2} \left (t \right )-16 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-36 x_{1} \left (t \right )-5 x_{2} \left (t \right )+16 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=72 x_{1} \left (t \right )+16 x_{2} \left (t \right )-29 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 66

ode:=[diff(x__1(t),t) = 39*x__1(t)+8*x__2(t)-16*x__3(t), diff(x__2(t),t) = -36*x__1(t)-5*x__2(t)+16*x__3(t), diff(x__3(t),t) = 72*x__1(t)+16*x__2(t)-29*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{3 t}+c_3 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= -c_2 \,{\mathrm e}^{3 t}-c_3 \,{\mathrm e}^{-t}+c_1 \,{\mathrm e}^{3 t} \\ x_{3} \left (t \right ) &= \frac {7 c_2 \,{\mathrm e}^{3 t}}{4}+2 c_3 \,{\mathrm e}^{-t}+\frac {c_1 \,{\mathrm e}^{3 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 127

ode={D[ x1[t],t]==39*x1[t]+8*x2[t]-16*x3[t],D[ x2[t],t]==-36*x1[t]-5*x2[t]+16*x3[t],D[ x3[t],t]==72*x1[t]+16*x2[t]-29*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-t} \left (c_1 \left (10 e^{4 t}-9\right )+2 (c_2-2 c_3) \left (e^{4 t}-1\right )\right ) \\ \text {x2}(t)\to e^{-t} \left (-9 c_1 \left (e^{4 t}-1\right )-c_2 \left (e^{4 t}-2\right )+4 c_3 \left (e^{4 t}-1\right )\right ) \\ \text {x3}(t)\to e^{-t} \left (18 c_1 \left (e^{4 t}-1\right )+4 c_2 \left (e^{4 t}-1\right )+c_3 \left (8-7 e^{4 t}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-39*x__1(t) - 8*x__2(t) + 16*x__3(t) + Derivative(x__1(t), t),0),Eq(36*x__1(t) + 5*x__2(t) - 16*x__3(t) + Derivative(x__2(t), t),0),Eq(-72*x__1(t) - 16*x__2(t) + 29*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {C_{1} e^{- t}}{2} - \left (\frac {2 C_{2}}{9} - \frac {4 C_{3}}{9}\right ) e^{3 t}, \ x^{2}{\left (t \right )} = - \frac {C_{1} e^{- t}}{2} + C_{2} e^{3 t}, \ x^{3}{\left (t \right )} = C_{1} e^{- t} + C_{3} e^{3 t}\right ] \]