9.4.21 problem problem 21

Internal problem ID [985]
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 21
Date solved : Tuesday, March 04, 2025 at 12:06:51 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=5 x_{1} \left (t \right )-6 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=4 x_{1} \left (t \right )-2 x_{2} \left (t \right )-4 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 53

ode:=[diff(x__1(t),t) = 5*x__1(t)-6*x__3(t), diff(x__2(t),t) = 2*x__1(t)-x__2(t)-2*x__3(t), diff(x__3(t),t) = 4*x__1(t)-2*x__2(t)-4*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 148

ode={D[ x1[t],t]==5*x1[t]+0*x2[t]-6*x3[t],D[ x2[t],t]==2*x1[t]-1*x2[t]-2*x3[t],D[ x3[t],t]==4*x1[t]-2*x2[t]-4*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-t} \left (c_1 \left (3 e^{2 t}-2\right )+6 \left (e^t-1\right ) \left (c_2 \left (e^t-1\right )-c_3 e^t\right )\right ) \\ \text {x2}(t)\to e^{-t} \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (-4 e^t+2 e^{2 t}+3\right )-2 c_3 e^t \left (e^t-1\right )\right ) \\ \text {x3}(t)\to e^{-t} \left (2 c_1 \left (e^{2 t}-1\right )+2 c_2 \left (-5 e^t+2 e^{2 t}+3\right )+c_3 e^t \left (5-4 e^t\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.138 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-5*x__1(t) + 6*x__3(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) + x__2(t) + 2*x__3(t) + Derivative(x__2(t), t),0),Eq(-4*x__1(t) + 2*x__2(t) + 4*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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