9.4.39 problem problem 50

Internal problem ID [1003]
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 50
Date solved : Tuesday, March 04, 2025 at 12:07:14 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=9 x_{1} \left (t \right )+13 x_{2} \left (t \right )-13 x_{6} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-14 x_{1} \left (t \right )+19 x_{2} \left (t \right )-10 x_{3} \left (t \right )-20 x_{4} \left (t \right )+10 x_{5} \left (t \right )+4 x_{6} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-30 x_{1} \left (t \right )+12 x_{2} \left (t \right )-7 x_{3} \left (t \right )-30 x_{4} \left (t \right )+12 x_{5} \left (t \right )+18 x_{6} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-12 x_{1} \left (t \right )+10 x_{2} \left (t \right )-10 x_{3} \left (t \right )-9 x_{4} \left (t \right )+10 x_{5} \left (t \right )+2 x_{6} \left (t \right )\\ \frac {d}{d t}x_{5} \left (t \right )&=6 x_{1} \left (t \right )+9 x_{2} \left (t \right )+6 x_{4} \left (t \right )+5 x_{5} \left (t \right )-15 x_{6} \left (t \right )\\ \frac {d}{d t}x_{6} \left (t \right )&=-14 x_{1} \left (t \right )+23 x_{2} \left (t \right )-10 x_{3} \left (t \right )-20 x_{4} \left (t \right )+10 x_{5} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.151 (sec). Leaf size: 134

ode:=[diff(x__1(t),t) = 9*x__1(t)+13*x__2(t)-13*x__6(t), diff(x__2(t),t) = -14*x__1(t)+19*x__2(t)-10*x__3(t)-20*x__4(t)+10*x__5(t)+4*x__6(t), diff(x__3(t),t) = -30*x__1(t)+12*x__2(t)-7*x__3(t)-30*x__4(t)+12*x__5(t)+18*x__6(t), diff(x__4(t),t) = -12*x__1(t)+10*x__2(t)-10*x__3(t)-9*x__4(t)+10*x__5(t)+2*x__6(t), diff(x__5(t),t) = 6*x__1(t)+9*x__2(t)+6*x__4(t)+5*x__5(t)-15*x__6(t), diff(x__6(t),t) = -14*x__1(t)+23*x__2(t)-10*x__3(t)-20*x__4(t)+10*x__5(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_5 \,{\mathrm e}^{-4 t}+c_6 \,{\mathrm e}^{9 t} \\ x_{2} \left (t \right ) &= c_6 \,{\mathrm e}^{9 t}+c_4 \,{\mathrm e}^{3 t}+{\mathrm e}^{-7 t} c_3 \\ x_{3} \left (t \right ) &= {\mathrm e}^{-7 t} c_3 +c_2 \,{\mathrm e}^{5 t}-{\mathrm e}^{11 t} c_1 \\ x_{4} \left (t \right ) &= {\mathrm e}^{11 t} c_1 +c_4 \,{\mathrm e}^{3 t}+{\mathrm e}^{-7 t} c_3 \\ x_{5} \left (t \right ) &= c_2 \,{\mathrm e}^{5 t}+{\mathrm e}^{11 t} c_1 +c_5 \,{\mathrm e}^{-4 t} \\ x_{6} \left (t \right ) &= c_6 \,{\mathrm e}^{9 t}+c_5 \,{\mathrm e}^{-4 t}+c_4 \,{\mathrm e}^{3 t}+{\mathrm e}^{-7 t} c_3 \\ \end{align*}

✓ Mathematica. Time used: 0.123 (sec). Leaf size: 1882

ode={D[ x1[t],t]==9*x1[t]+13*x2[t]-13*x6[t],D[ x2[t],t]==-14*x1[t]+19*x2[t]-10*x3[t]-20*x4[t]+10*x5[t]+4*x6[t],D[ x3[t],t]==-30*x1[t]+12*x2[t]-7*x3[t]-30*x4[t]+12*x5[t]+18*x6[t],D[ x4[t],t]==-12*x1[t]+10*x2[t]-10*x3[t]-9*x4[t]+10*x5[t]+2*x6[t],D[ x5[t],t]==6*x1[t]+9*x2[t]+6*x4[t]+5*x5[t]-15*x6[t],D[ x6[t],t]==-14*x1[t]+23*x2[t]-10*x3[t]-20*x4[t]-10*x5[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t],x5[t],x6[t]},t,IncludeSingularSolutions->True]
 

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✓ Sympy. Time used: 0.319 (sec). Leaf size: 133

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
x__4 = Function("x__4") 
x__5 = Function("x__5") 
x__6 = Function("x__6") 
ode=[Eq(-9*x__1(t) - 13*x__2(t) + 13*x__6(t) + Derivative(x__1(t), t),0),Eq(14*x__1(t) - 19*x__2(t) + 10*x__3(t) + 20*x__4(t) - 10*x__5(t) - 4*x__6(t) + Derivative(x__2(t), t),0),Eq(30*x__1(t) - 12*x__2(t) + 7*x__3(t) + 30*x__4(t) - 12*x__5(t) - 18*x__6(t) + Derivative(x__3(t), t),0),Eq(12*x__1(t) - 10*x__2(t) + 10*x__3(t) + 9*x__4(t) - 10*x__5(t) - 2*x__6(t) + Derivative(x__4(t), t),0),Eq(-6*x__1(t) - 9*x__2(t) - 6*x__4(t) - 5*x__5(t) + 15*x__6(t) + Derivative(x__5(t), t),0),Eq(14*x__1(t) - 23*x__2(t) + 10*x__3(t) + 20*x__4(t) - 10*x__5(t) + Derivative(x__6(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t),x__5(t),x__6(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{9 t}, \ x^{2}{\left (t \right )} = C_{2} e^{9 t} + C_{3} e^{- 7 t} + C_{4} e^{3 t}, \ x^{3}{\left (t \right )} = C_{3} e^{- 7 t} + C_{5} e^{5 t} - C_{6} e^{11 t}, \ x^{4}{\left (t \right )} = C_{3} e^{- 7 t} + C_{4} e^{3 t} + C_{6} e^{11 t}, \ x^{5}{\left (t \right )} = C_{1} e^{- 4 t} + C_{5} e^{5 t} + C_{6} e^{11 t}, \ x^{6}{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{9 t} + C_{3} e^{- 7 t} + C_{4} e^{3 t}\right ] \]