9.6.13 problem problem 13

Internal problem ID [1020]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 13
Date solved : Tuesday, March 04, 2025 at 12:07:34 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 53

ode:=[diff(x__1(t),t) = -x__1(t)+x__3(t), diff(x__2(t),t) = x__2(t)-4*x__3(t), diff(x__3(t),t) = x__2(t)-3*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 78

ode={D[ x1[t],t]==-1*x1[t]+0*x2[t]+1*x3[t],D[ x2[t],t]==0*x1[t]+1*x2[t]-4*x3[t],D[ x3[t],t]==0*x1[t]+1*x2[t]-3*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.116 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(x__1(t) - x__3(t) + Derivative(x__1(t), t),0),Eq(-x__2(t) + 4*x__3(t) + Derivative(x__2(t), t),0),Eq(-x__2(t) + 3*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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