9.6.26 problem problem 26

Internal problem ID [1033]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 26
Date solved : Tuesday, March 04, 2025 at 12:07:46 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 61

ode:=[diff(x__1(t),t) = 5*x__1(t)-x__2(t)+x__3(t), diff(x__2(t),t) = x__1(t)+3*x__2(t), diff(x__3(t),t) = -3*x__1(t)+2*x__2(t)+x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 105

ode={D[ x1[t],t]==5*x1[t]-1*x2[t]+1*x3[t],D[ x2[t],t]==1*x1[t]+3*x2[t]+0*x3[t],D[ x3[t],t]==-3*x1[t]+2*x2[t]+1*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.173 (sec). Leaf size: 94

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) + x__2(t) - x__3(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) - 3*x__2(t) + Derivative(x__2(t), t),0),Eq(3*x__1(t) - 2*x__2(t) - x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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