9.4.28 problem problem 39

Internal problem ID [992]
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 39
Date solved : Tuesday, March 04, 2025 at 12:06:59 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-2 x_{1} \left (t \right )+9 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )+2 x_{2} \left (t \right )-10 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{3} \left (t \right )+8 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 60

ode:=[diff(x__1(t),t) = -2*x__1(t)+9*x__4(t), diff(x__2(t),t) = 4*x__1(t)+2*x__2(t)-10*x__4(t), diff(x__3(t),t) = -x__3(t)+8*x__4(t), diff(x__4(t),t) = x__4(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 103

ode={D[ x1[t],t]==-2*x1[t]+0*x2[t]+0*x3[t]+9*x4[t],D[ x2[t],t]==4*x1[t]+2*x2[t]+0*x3[t]-10*x4[t],D[ x3[t],t]==0*x1[t]+0*x2[t]-1*x3[t]+8*x4[t],D[ x4[t],t]==0*x1[t]+0*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^{-2 t} \left (3 c_4 \left (e^{3 t}-1\right )+c_1\right ) \\ \text {x2}(t)\to e^{-2 t} \left (c_1 \left (e^{4 t}-1\right )+(c_2-c_4) e^{4 t}-2 c_4 e^{3 t}+3 c_4\right ) \\ \text {x3}(t)\to e^{-t} \left (4 c_4 \left (e^{2 t}-1\right )+c_3\right ) \\ \text {x4}(t)\to c_4 e^t \\ \end{align*}

✓ Sympy. Time used: 0.156 (sec). Leaf size: 58

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) - 9*x__4(t) + Derivative(x__1(t), t),0),Eq(-4*x__1(t) - 2*x__2(t) + 10*x__4(t) + Derivative(x__2(t), t),0),Eq(x__3(t) - 8*x__4(t) + Derivative(x__3(t), t),0),Eq(-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)
 

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