9.4.26 problem problem 26

Internal problem ID [990]
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 26
Date solved : Tuesday, March 04, 2025 at 12:06:56 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=9 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 )&=-9 x_{1} \left (t \right )+4 x_{2} \left (t \right )-x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 0\\ x_{3} \left (0\right ) = 17 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 63

ode:=[diff(x__1(t),t) = 3*x__1(t)+x__3(t), diff(x__2(t),t) = 9*x__1(t)-x__2(t)+2*x__3(t), diff(x__3(t),t) = -9*x__1(t)+4*x__2(t)-x__3(t)]; 
ic:=x__1(0) = 0x__2(0) = 0x__3(0) = 17; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= 4 \,{\mathrm e}^{3 t}+{\mathrm e}^{-t} \sin \left (t \right )-4 \,{\mathrm e}^{-t} \cos \left (t \right ) \\ x_{2} \left (t \right ) &= 9 \,{\mathrm e}^{3 t}-9 \,{\mathrm e}^{-t} \cos \left (t \right )-2 \,{\mathrm e}^{-t} \sin \left (t \right ) \\ x_{3} \left (t \right ) &= 17 \,{\mathrm e}^{-t} \cos \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 62

ode={D[ x1[t],t]==3*x1[t]+0*x2[t]+1*x3[t],D[ x2[t],t]==9*x1[t]-1*x2[t]+2*x3[t],D[ x3[t],t]==-9*x1[t]+4*x2[t]-1*x3[t]}; 
ic={x1[0]==0,x2[0]==0,x3[0]==17}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-t} \left (4 e^{4 t}+\sin (t)-4 \cos (t)\right ) \\ \text {x2}(t)\to e^{-t} \left (9 e^{4 t}-2 \sin (t)-9 \cos (t)\right ) \\ \text {x3}(t)\to 17 e^{-t} \cos (t) \\ \end{align*}

✓ Sympy. Time used: 0.184 (sec). Leaf size: 104

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-3*x__1(t) - x__3(t) + Derivative(x__1(t), t),0),Eq(-9*x__1(t) + x__2(t) - 2*x__3(t) + Derivative(x__2(t), t),0),Eq(9*x__1(t) - 4*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)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {4 C_{3} e^{3 t}}{9} + \left (\frac {C_{1}}{17} + \frac {4 C_{2}}{17}\right ) e^{- t} \sin {\left (t \right )} - \left (\frac {4 C_{1}}{17} - \frac {C_{2}}{17}\right ) e^{- t} \cos {\left (t \right )}, \ x^{2}{\left (t \right )} = C_{3} e^{3 t} - \left (\frac {2 C_{1}}{17} - \frac {9 C_{2}}{17}\right ) e^{- t} \sin {\left (t \right )} - \left (\frac {9 C_{1}}{17} + \frac {2 C_{2}}{17}\right ) e^{- t} \cos {\left (t \right )}, \ x^{3}{\left (t \right )} = C_{1} e^{- t} \cos {\left (t \right )} - C_{2} e^{- t} \sin {\left (t \right )}\right ] \]