9.4.33 problem problem 44

Internal problem ID [997]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 44
Date solved : Tuesday, March 04, 2025 at 12:07:05 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=147 x_{1} \left (t \right )+23 x_{2} \left (t \right )-202 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-90 x_{1} \left (t \right )-9 x_{2} \left (t \right )+129 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=90 x_{1} \left (t \right )+15 x_{2} \left (t \right )-123 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 73

ode:=[diff(x__1(t),t) = 147*x__1(t)+23*x__2(t)-202*x__3(t), diff(x__2(t),t) = -90*x__1(t)-9*x__2(t)+129*x__3(t), diff(x__3(t),t) = 90*x__1(t)+15*x__2(t)-123*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{6 t}+c_2 \,{\mathrm e}^{-3 t}+c_3 \,{\mathrm e}^{12 t} \\ x_{2} \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{6 t}}{7}-\frac {2 c_2 \,{\mathrm e}^{-3 t}}{3}-\frac {3 c_3 \,{\mathrm e}^{12 t}}{5} \\ x_{3} \left (t \right ) &= \frac {5 c_1 \,{\mathrm e}^{6 t}}{7}+\frac {2 c_2 \,{\mathrm e}^{-3 t}}{3}+\frac {3 c_3 \,{\mathrm e}^{12 t}}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 188

ode={D[ x1[t],t]==147*x1[t]+23*x2[t]-202*x3[t],D[ x2[t],t]==-90*x1[t]-9*x2[t]+129*x3[t],D[ x3[t],t]==90*x1[t]+15*x2[t]-123*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{6} e^{-3 t} \left (6 c_1 \left (10 e^{15 t}-9\right )+c_2 \left (7 e^{9 t}+5 e^{15 t}-12\right )-c_3 \left (-7 e^{9 t}+85 e^{15 t}-78\right )\right ) \\ \text {x2}(t)\to \frac {1}{6} e^{-3 t} \left (-36 c_1 \left (e^{15 t}-1\right )+c_2 \left (e^{9 t}-3 e^{15 t}+8\right )+c_3 \left (e^{9 t}+51 e^{15 t}-52\right )\right ) \\ \text {x3}(t)\to \frac {1}{6} e^{-3 t} \left (36 c_1 \left (e^{15 t}-1\right )+c_2 \left (5 e^{9 t}+3 e^{15 t}-8\right )-c_3 \left (-5 e^{9 t}+51 e^{15 t}-52\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.176 (sec). Leaf size: 78

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-147*x__1(t) - 23*x__2(t) + 202*x__3(t) + Derivative(x__1(t), t),0),Eq(90*x__1(t) + 9*x__2(t) - 129*x__3(t) + Derivative(x__2(t), t),0),Eq(-90*x__1(t) - 15*x__2(t) + 123*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {3 C_{1} e^{- 3 t}}{2} + \frac {7 C_{2} e^{6 t}}{5} + \frac {5 C_{3} e^{12 t}}{3}, \ x^{2}{\left (t \right )} = - C_{1} e^{- 3 t} + \frac {C_{2} e^{6 t}}{5} - C_{3} e^{12 t}, \ x^{3}{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{6 t} + C_{3} e^{12 t}\right ] \]