9.6.25 problem problem 25

Internal problem ID [1032]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 25
Date solved : Saturday, March 29, 2025 at 10:37:18 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-2 x_{1} \left (t \right )+17 x_{2} \left (t \right )+4 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )+6 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{2} \left (t \right )+2 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.124 (sec). Leaf size: 61

ode:=[diff(x__1(t),t) = -2*x__1(t)+17*x__2(t)+4*x__3(t), diff(x__2(t),t) = -x__1(t)+6*x__2(t)+x__3(t), diff(x__3(t),t) = x__2(t)+2*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (c_3 \,t^{2}+c_2 t +8 c_3 t +c_1 +4 c_2 -2 c_3 \right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{2 t} \left (2 c_3 t +c_2 \right ) \\ x_{3} \left (t \right ) &= {\mathrm e}^{2 t} \left (c_3 \,t^{2}+c_2 t +c_1 \right ) \\ \end{align*}

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

ode={D[ x1[t],t]==-2*x1[t]+17*x2[t]+4*x3[t],D[ x2[t],t]==-1*x1[t]+6*x2[t]+1*x3[t],D[ x3[t],t]==0*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^{2 t} \left (-\left (c_1 \left (t^2+8 t-2\right )\right )+c_2 t (4 t+34)+c_3 t (t+8)\right ) \\ \text {x2}(t)\to e^{2 t} ((-c_1+4 c_2+c_3) t+c_2) \\ \text {x3}(t)\to \frac {1}{2} e^{2 t} \left ((-c_1+4 c_2+c_3) t^2+2 c_2 t+2 c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.175 (sec). Leaf size: 88

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(2*x__1(t) - 17*x__2(t) - 4*x__3(t) + Derivative(x__1(t), t),0),Eq(x__1(t) - 6*x__2(t) - x__3(t) + Derivative(x__2(t), t),0),Eq(-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)
 

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