76.26.4 problem 4
Internal
problem
ID
[17765]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
6.
Systems
of
First
Order
Linear
Equations.
Section
6.4
(Nondefective
Matrices
with
Complex
Eigenvalues).
Problems
at
page
419
Problem
number
:
4
Date
solved
:
Monday, March 31, 2025 at 04:27:02 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-4 x_{1} \left (t \right )+2 x_{2} \left (t \right )-x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-6 x_{1} \left (t \right )-3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=\frac {8 x_{2} \left (t \right )}{3}-2 x_{3} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.304 (sec). Leaf size: 96
ode:=[diff(x__1(t),t) = -4*x__1(t)+2*x__2(t)-x__3(t), diff(x__2(t),t) = -6*x__1(t)-3*x__3(t), diff(x__3(t),t) = 8/3*x__2(t)-2*x__3(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= \frac {{\mathrm e}^{-2 t} \left (2 \sin \left (4 t \right ) c_2 +2 c_3 \sin \left (4 t \right )-2 c_2 \cos \left (4 t \right )+2 \cos \left (4 t \right ) c_3 -3 c_1 -2 c_2 \right )}{6} \\
x_{2} \left (t \right ) &= {\mathrm e}^{-2 t} \left (\sin \left (4 t \right ) c_2 +\cos \left (4 t \right ) c_3 \right ) \\
x_{3} \left (t \right ) &= \frac {{\mathrm e}^{-2 t} \left (2 c_3 \sin \left (4 t \right )-2 c_2 \cos \left (4 t \right )+3 c_1 +2 c_2 \right )}{3} \\
\end{align*}
✓ Mathematica. Time used: 0.009 (sec). Leaf size: 152
ode={D[x1[t],t]==-4*x1[t]+2*x2[t]-1*x3[t],D[x2[t],t]==-6*x1[t]+0*x2[t]-3*x3[t],D[x3[t],t]==0*x1[t]+8/3*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}{12} e^{-2 t} ((6 c_1+2 c_2+3 c_3) \cos (4 t)-3 (2 c_1-2 c_2+c_3) \sin (4 t)+6 c_1-2 c_2-3 c_3) \\
\text {x2}(t)\to \frac {1}{4} e^{-2 t} (4 c_2 \cos (4 t)+(-6 c_1+2 c_2-3 c_3) \sin (4 t)) \\
\text {x3}(t)\to \frac {1}{6} e^{-2 t} ((6 c_1-2 c_2+3 c_3) \cos (4 t)+4 c_2 \sin (4 t)-6 c_1+2 c_2+3 c_3) \\
\end{align*}
✓ Sympy. Time used: 0.169 (sec). Leaf size: 110
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) - 2*x__2(t) + x__3(t) + Derivative(x__1(t), t),0),Eq(6*x__1(t) + 3*x__3(t) + Derivative(x__2(t), t),0),Eq(-8*x__2(t)/3 + 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)
\[
\left [ x^{1}{\left (t \right )} = - \frac {C_{1} e^{- 2 t}}{2} + \left (\frac {C_{2}}{2} - \frac {C_{3}}{2}\right ) e^{- 2 t} \cos {\left (4 t \right )} - \left (\frac {C_{2}}{2} + \frac {C_{3}}{2}\right ) e^{- 2 t} \sin {\left (4 t \right )}, \ x^{2}{\left (t \right )} = - \frac {3 C_{2} e^{- 2 t} \sin {\left (4 t \right )}}{2} - \frac {3 C_{3} e^{- 2 t} \cos {\left (4 t \right )}}{2}, \ x^{3}{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{- 2 t} \cos {\left (4 t \right )} - C_{3} e^{- 2 t} \sin {\left (4 t \right )}\right ]
\]