20.20.12 problem 12
Internal
problem
ID
[3902]
Book
:
Differential
equations
and
linear
algebra,
Stephen
W.
Goode
and
Scott
A
Annin.
Fourth
edition,
2015
Section
:
Chapter
9,
First
order
linear
systems.
Section
9.11
(Chapter
review),
page
665
Problem
number
:
12
Date
solved
:
Tuesday, March 04, 2025 at 05:19:06 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-4 x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+5 x_{2} \left (t \right )-9 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=5 x_{2} \left (t \right )-x_{3} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.242 (sec). Leaf size: 96
ode:=[diff(x__1(t),t) = -4*x__1(t), diff(x__2(t),t) = 2*x__1(t)+5*x__2(t)-9*x__3(t), diff(x__3(t),t) = 5*x__2(t)-x__3(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= c_3 \,{\mathrm e}^{-4 t} \\
x_{2} \left (t \right ) &= -\frac {c_3 \,{\mathrm e}^{-4 t}}{12}-\frac {6 \,{\mathrm e}^{2 t} \sin \left (6 t \right ) c_{1}}{5}+\frac {6 \,{\mathrm e}^{2 t} \cos \left (6 t \right ) c_{2}}{5}+\frac {3 \,{\mathrm e}^{2 t} \sin \left (6 t \right ) c_{2}}{5}+\frac {3 \,{\mathrm e}^{2 t} \cos \left (6 t \right ) c_{1}}{5} \\
x_{3} \left (t \right ) &= \frac {5 c_3 \,{\mathrm e}^{-4 t}}{36}+{\mathrm e}^{2 t} \sin \left (6 t \right ) c_{2} +{\mathrm e}^{2 t} \cos \left (6 t \right ) c_{1} \\
\end{align*}
✓ Mathematica. Time used: 0.009 (sec). Leaf size: 129
ode={D[x1[t],t]==-4*x1[t]-0*x2[t]+0*x3[t],D[x2[t],t]==2*x1[t]+5*x2[t]-9*x3[t],D[x3[t],t]==0*x1[t]+5*x2[t]-1*x3[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to c_1 e^{-4 t} \\
\text {x2}(t)\to \frac {1}{12} e^{-4 t} \left ((c_1+12 c_2) e^{6 t} \cos (6 t)+3 (c_1+2 c_2-6 c_3) e^{6 t} \sin (6 t)-c_1\right ) \\
\text {x3}(t)\to \frac {1}{36} e^{-4 t} \left (-(5 c_1-36 c_3) e^{6 t} \cos (6 t)+(5 c_1+30 c_2-18 c_3) e^{6 t} \sin (6 t)+5 c_1\right ) \\
\end{align*}
✓ Sympy. Time used: 0.188 (sec). Leaf size: 97
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
x__3 = Function("x__3")
ode=[Eq(4*x__1(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - 5*x__2(t) + 9*x__3(t) + Derivative(x__2(t), t),0),Eq(-5*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)
\[
\left [ x^{1}{\left (t \right )} = \frac {36 C_{1} e^{- 4 t}}{5}, \ x^{2}{\left (t \right )} = - \frac {3 C_{1} e^{- 4 t}}{5} + \left (\frac {3 C_{2}}{5} - \frac {6 C_{3}}{5}\right ) e^{2 t} \cos {\left (6 t \right )} - \left (\frac {6 C_{2}}{5} + \frac {3 C_{3}}{5}\right ) e^{2 t} \sin {\left (6 t \right )}, \ x^{3}{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{2 t} \cos {\left (6 t \right )} - C_{3} e^{2 t} \sin {\left (6 t \right )}\right ]
\]