9.6.14 problem problem 14

Internal problem ID [1021]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 14
Date solved : Saturday, March 29, 2025 at 10:37:03 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.114 (sec). Leaf size: 71

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 119

ode={D[ x1[t],t]==0*x1[t]+0*x2[t]+1*x3[t],D[ x2[t],t]==-5*x1[t]-1*x2[t]-5*x3[t],D[ x3[t],t]==4*x1[t]+1*x2[t]-2*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} \left (c_1 \left (5 t^2+2 t+2\right )+t (c_2 t+2 c_3)\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (-5 (5 c_1+c_2) t^2-10 (c_1+c_3) t+2 c_2\right ) \\ \text {x3}(t)\to \frac {1}{2} e^{-t} \left (-\left ((5 c_1+c_2) t^2\right )+2 (4 c_1+c_2-c_3) t+2 c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.160 (sec). Leaf size: 107

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

\[ \left [ x^{1}{\left (t \right )} = \frac {5 C_{2} t^{2} e^{- t}}{2} + t \left (5 C_{1} + C_{2}\right ) e^{- t} + \left (C_{1} + C_{2} + 5 C_{3}\right ) e^{- t}, \ x^{2}{\left (t \right )} = - \frac {25 C_{2} t^{2} e^{- t}}{2} - t \left (25 C_{1} + 5 C_{2}\right ) e^{- t} - \left (5 C_{1} + 25 C_{3}\right ) e^{- t}, \ x^{3}{\left (t \right )} = - \frac {5 C_{2} t^{2} e^{- t}}{2} - t \left (5 C_{1} - 4 C_{2}\right ) e^{- t} + \left (4 C_{1} - 5 C_{3}\right ) e^{- t}\right ] \]