9.4.32 problem problem 43

Internal problem ID [996]
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 43
Date solved : Tuesday, March 04, 2025 at 12:07:04 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-20 x_{1} \left (t \right )+11 x_{2} \left (t \right )+13 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=12 x_{1} \left (t \right )-x_{2} \left (t \right )-7 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-48 x_{1} \left (t \right )+21 x_{2} \left (t \right )+31 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 71

ode:=[diff(x__1(t),t) = -20*x__1(t)+11*x__2(t)+13*x__3(t), diff(x__2(t),t) = 12*x__1(t)-x__2(t)-7*x__3(t), diff(x__3(t),t) = -48*x__1(t)+21*x__2(t)+31*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-2 t}+c_2 \,{\mathrm e}^{8 t}+c_3 \,{\mathrm e}^{4 t} \\ x_{2} \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{-2 t}}{3}-c_2 \,{\mathrm e}^{8 t}+c_3 \,{\mathrm e}^{4 t} \\ x_{3} \left (t \right ) &= \frac {5 c_1 \,{\mathrm e}^{-2 t}}{3}+3 c_2 \,{\mathrm e}^{8 t}+c_3 \,{\mathrm e}^{4 t} \\ \end{align*}

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 554

ode={D[ x1[t],t]==20*x1[t]+11*x2[t]+13*x3[t],D[ x2[t],t]==12*x1[t]-1*x2[t]-7*x3[t],D[ x3[t],t]==-48*x1[t]+21*x2[t]+31*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 0.167 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(20*x__1(t) - 11*x__2(t) - 13*x__3(t) + Derivative(x__1(t), t),0),Eq(-12*x__1(t) + x__2(t) + 7*x__3(t) + Derivative(x__2(t), t),0),Eq(48*x__1(t) - 21*x__2(t) - 31*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {3 C_{1} e^{- 2 t}}{5} + C_{2} e^{4 t} + \frac {C_{3} e^{8 t}}{3}, \ x^{2}{\left (t \right )} = - \frac {C_{1} e^{- 2 t}}{5} + C_{2} e^{4 t} - \frac {C_{3} e^{8 t}}{3}, \ x^{3}{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{4 t} + C_{3} e^{8 t}\right ] \]