20.13.8 problem 8
Internal
problem
ID
[3817]
Book
:
Differential
equations
and
linear
algebra,
Stephen
W.
Goode
and
Scott
A
Annin.
Fourth
edition,
2015
Section
:
Chapter
9,
First
order
linear
systems.
Section
9.1,
page
587
Problem
number
:
8
Date
solved
:
Tuesday, March 04, 2025 at 05:17:28 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-2 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 )-x_{2} \left (t \right )+3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{2} \left (t \right )-3 x_{3} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.051 (sec). Leaf size: 68
ode:=[diff(x__1(t),t) = -2*x__1(t)+x__2(t)+x__3(t), diff(x__2(t),t) = x__1(t)-x__2(t)+3*x__3(t), diff(x__3(t),t) = -x__2(t)-3*x__3(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= -{\mathrm e}^{-2 t} \left (\sin \left (t \right ) c_{2} +\cos \left (t \right ) c_3 +2 c_{1} \right ) \\
x_{2} \left (t \right ) &= -{\mathrm e}^{-2 t} \left (\sin \left (t \right ) c_{2} -c_3 \sin \left (t \right )+c_{2} \cos \left (t \right )+\cos \left (t \right ) c_3 +c_{1} \right ) \\
x_{3} \left (t \right ) &= {\mathrm e}^{-2 t} \left (c_{1} +\sin \left (t \right ) c_{2} +\cos \left (t \right ) c_3 \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.011 (sec). Leaf size: 114
ode={D[x1[t],t]==-2*x1[t]+x2[t]+x3[t],D[x2[t],t]==x1[t]-x2[t]+3*x3[t],D[x3[t],t]==-x2[t]-3*x3[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to e^{-2 t} (-(c_1+2 c_3) \cos (t)+(c_2+c_3) \sin (t)+2 (c_1+c_3)) \\
\text {x2}(t)\to e^{-2 t} (-(c_1-c_2+c_3) \cos (t)+(c_1+c_2+3 c_3) \sin (t)+c_1+c_3) \\
\text {x3}(t)\to e^{-2 t} ((c_1+2 c_3) \cos (t)-(c_2+c_3) \sin (t)-c_1-c_3) \\
\end{align*}
✓ Sympy. Time used: 0.160 (sec). Leaf size: 92
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) - x__2(t) - x__3(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + x__2(t) - 3*x__3(t) + Derivative(x__2(t), t),0),Eq(x__2(t) + 3*x__3(t) + Derivative(x__3(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - 2 C_{1} e^{- 2 t} + C_{2} e^{- 2 t} \sin {\left (t \right )} - C_{3} e^{- 2 t} \cos {\left (t \right )}, \ x^{2}{\left (t \right )} = - C_{1} e^{- 2 t} + \left (C_{2} - C_{3}\right ) e^{- 2 t} \cos {\left (t \right )} + \left (C_{2} + C_{3}\right ) e^{- 2 t} \sin {\left (t \right )}, \ x^{3}{\left (t \right )} = C_{1} e^{- 2 t} - C_{2} e^{- 2 t} \sin {\left (t \right )} + C_{3} e^{- 2 t} \cos {\left (t \right )}\right ]
\]