9.4.29 problem problem 40

Internal problem ID [993]
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 40
Date solved : Tuesday, March 04, 2025 at 12:07:00 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-21 x_{1} \left (t \right )-5 x_{2} \left (t \right )-27 x_{3} \left (t \right )-9 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=5 x_{3} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-21 x_{3} \left (t \right )-2 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 60

ode:=[diff(x__1(t),t) = 2*x__1(t), diff(x__2(t),t) = -21*x__1(t)-5*x__2(t)-27*x__3(t)-9*x__4(t), diff(x__3(t),t) = 5*x__3(t), diff(x__4(t),t) = -21*x__3(t)-2*x__4(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 86

ode={D[ x1[t],t]==2*x1[t]+0*x2[t]+0*x3[t]+0*x4[t],D[ x2[t],t]==-21*x1[t]-5*x2[t]-27*x3[t]-9*x4[t],D[ x3[t],t]==0*x1[t]+0*x2[t]+5*x3[t]+0*x4[t],D[ x4[t],t]==0*x1[t]+0*x2[t]-21*x3[t]-2*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to c_1 e^{2 t} \\ \text {x2}(t)\to e^{-5 t} \left (-3 c_1 \left (e^{7 t}-1\right )-3 (3 c_3+c_4) \left (e^{3 t}-1\right )+c_2\right ) \\ \text {x3}(t)\to c_3 e^{5 t} \\ \text {x4}(t)\to e^{-2 t} \left (c_4-3 c_3 \left (e^{7 t}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.155 (sec). Leaf size: 63

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") 
ode=[Eq(-2*x__1(t) + Derivative(x__1(t), t),0),Eq(21*x__1(t) + 5*x__2(t) + 27*x__3(t) + 9*x__4(t) + Derivative(x__2(t), t),0),Eq(-5*x__3(t) + Derivative(x__3(t), t),0),Eq(21*x__3(t) + 2*x__4(t) + Derivative(x__4(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
 

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