9.4.38 problem problem 49

Internal problem ID [1002]
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 49
Date solved : Tuesday, March 04, 2025 at 12:07:12 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=139 x_{1} \left (t \right )-14 x_{2} \left (t \right )-52 x_{3} \left (t \right )-14 x_{4} \left (t \right )+28 x_{5} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-22 x_{1} \left (t \right )+5 x_{2} \left (t \right )+7 x_{3} \left (t \right )+8 x_{4} \left (t \right )-7 x_{5} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=370 x_{1} \left (t \right )-38 x_{2} \left (t \right )-139 x_{3} \left (t \right )-38 x_{4} \left (t \right )+76 x_{5} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=152 x_{1} \left (t \right )-16 x_{2} \left (t \right )-59 x_{3} \left (t \right )-13 x_{4} \left (t \right )+35 x_{5} \left (t \right )\\ \frac {d}{d t}x_{5} \left (t \right )&=95 x_{1} \left (t \right )-10 x_{2} \left (t \right )-38 x_{3} \left (t \right )-7 x_{4} \left (t \right )+23 x_{5} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 131

ode:=[diff(x__1(t),t) = 139*x__1(t)-14*x__2(t)-52*x__3(t)-14*x__4(t)+28*x__5(t), diff(x__2(t),t) = -22*x__1(t)+5*x__2(t)+7*x__3(t)+8*x__4(t)-7*x__5(t), diff(x__3(t),t) = 370*x__1(t)-38*x__2(t)-139*x__3(t)-38*x__4(t)+76*x__5(t), diff(x__4(t),t) = 152*x__1(t)-16*x__2(t)-59*x__3(t)-13*x__4(t)+35*x__5(t), diff(x__5(t),t) = 95*x__1(t)-10*x__2(t)-38*x__3(t)-7*x__4(t)+23*x__5(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 2676

ode={D[ x1[t],t]==139*x1[t]-14*x2[t]-52*x3[t]-14*x4[t]+28*x5[t],D[ x2[t],t]==-22*x1[t]+5*x2[t]+7*x3[t]+8*x4[t]-7*x5[t],D[ x3[t],t]==370*x1[t]-38*x2[t]-139*x3[t]-38*x4[t]+76*x5[t],D[ x4[t],t]==152*x1[t]-16*x2[t]-59*x3[t]-13*x4[t]+45*x5[t],D[ x5[t],t]==95*x1[t]-10*x2[t]-38*x3[t]-7*x4[t]+23*x5[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t],x5[t]},t,IncludeSingularSolutions->True]
 

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✓ Sympy. Time used: 0.306 (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") 
ode=[Eq(-139*x__1(t) + 14*x__2(t) + 52*x__3(t) + 14*x__4(t) - 28*x__5(t) + Derivative(x__1(t), t),0),Eq(22*x__1(t) - 5*x__2(t) - 7*x__3(t) - 8*x__4(t) + 7*x__5(t) + Derivative(x__2(t), t),0),Eq(-370*x__1(t) + 38*x__2(t) + 139*x__3(t) + 38*x__4(t) - 76*x__5(t) + Derivative(x__3(t), t),0),Eq(-152*x__1(t) + 16*x__2(t) + 59*x__3(t) + 13*x__4(t) - 35*x__5(t) + Derivative(x__4(t), t),0),Eq(-95*x__1(t) + 10*x__2(t) + 38*x__3(t) + 7*x__4(t) - 23*x__5(t) + Derivative(x__5(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t),x__5(t)],ics=ics)
 

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