9.6.19 problem problem 19

Internal problem ID [1026]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 19
Date solved : Saturday, March 29, 2025 at 10:37:09 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-4 x_{2} \left (t \right )-2 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=6 x_{1} \left (t \right )-12 x_{2} \left (t \right )-x_{3} \left (t \right )-6 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-4 x_{2} \left (t \right )-x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.180 (sec). Leaf size: 52

ode:=[diff(x__1(t),t) = x__1(t)-4*x__2(t)-2*x__4(t), diff(x__2(t),t) = x__2(t), diff(x__3(t),t) = 6*x__1(t)-12*x__2(t)-x__3(t)-6*x__4(t), diff(x__4(t),t) = -4*x__2(t)-x__4(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 114

ode={D[ x1[t],t]==1*x1[t]-4*x2[t]+0*x3[t]-2*x4[t],D[ x2[t],t]==0*x1[t]+1*x2[t]+0*x3[t]+0*x4[t],D[ x3[t],t]==6*x1[t]-12*x2[t]-1*x3[t]-6*x4[t],D[ x4[t],t]==0*x1[t]-4*x2[t]+0*x3[t]-1*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-t} \left ((c_1-2 c_2-c_4) e^{2 t}+2 c_2+c_4\right ) \\ \text {x2}(t)\to c_2 e^t \\ \text {x3}(t)\to e^{-t} \left (3 c_1 \left (e^{2 t}-1\right )-6 c_2 \left (e^{2 t}-1\right )-3 c_4 e^{2 t}+c_3+3 c_4\right ) \\ \text {x4}(t)\to e^{-t} \left (c_4-2 c_2 \left (e^{2 t}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.133 (sec). Leaf size: 48

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(-x__1(t) + 4*x__2(t) + 2*x__4(t) + Derivative(x__1(t), t),0),Eq(-x__2(t) + Derivative(x__2(t), t),0),Eq(-6*x__1(t) + 12*x__2(t) + x__3(t) + 6*x__4(t) + Derivative(x__3(t), t),0),Eq(4*x__2(t) + 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)
 

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