20.19.10 problem 11
Internal
problem
ID
[3890]
Book
:
Differential
equations
and
linear
algebra,
Stephen
W.
Goode
and
Scott
A
Annin.
Fourth
edition,
2015
Section
:
Chapter
9,
First
order
linear
systems.
Section
9.8
(Matrix
exponential
function),
page
642
Problem
number
:
11
Date
solved
:
Sunday, March 30, 2025 at 02:11:09 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{2} \left (t \right )-x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=x_{2} \left (t \right )+x_{3} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.412 (sec). Leaf size: 123
ode:=[diff(x__1(t),t) = -x__2(t), diff(x__2(t),t) = x__1(t), diff(x__3(t),t) = x__2(t)-x__4(t), diff(x__4(t),t) = x__2(t)+x__3(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= c_3 \sin \left (t \right )+c_4 \cos \left (t \right ) \\
x_{2} \left (t \right ) &= -c_3 \cos \left (t \right )+c_4 \sin \left (t \right ) \\
x_{3} \left (t \right ) &= \sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 +\frac {c_3 \cos \left (t \right )}{2}+\frac {\sin \left (t \right ) c_3 t}{2}+\frac {c_4 \cos \left (t \right )}{2}+\frac {\sin \left (t \right ) c_4 t}{2}-\frac {\cos \left (t \right ) c_3 t}{2}+\frac {\cos \left (t \right ) c_4 t}{2} \\
x_{4} \left (t \right ) &= -\cos \left (t \right ) c_2 +\sin \left (t \right ) c_1 -\frac {\cos \left (t \right ) c_3 t}{2}-\frac {\cos \left (t \right ) c_4 t}{2}-\frac {c_3 \cos \left (t \right )}{2}-\frac {c_4 \cos \left (t \right )}{2}-\frac {\sin \left (t \right ) c_3 t}{2}+\frac {\sin \left (t \right ) c_4 t}{2}+c_4 \sin \left (t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.031 (sec). Leaf size: 80
ode={D[x1[t],t]==0*x1[t]-1*x2[t]+0*x3[t]+0*x4[t],D[x2[t],t]==1*x1[t]+0*x2[t]-0*x3[t]+0*x4[t],D[x3[t],t]==1*x1[t]+0*x2[t]+0*x3[t]-x4[t],D[x4[t],t]==0*x1[t]+1*x2[t]+1*x3[t]-0*x4[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to c_1 \cos (t)-c_2 \sin (t) \\
\text {x2}(t)\to c_2 \cos (t)+c_1 \sin (t) \\
\text {x3}(t)\to (c_1 t+c_3) \cos (t)-(c_2 t+c_4) \sin (t) \\
\text {x4}(t)\to (c_2 t+c_4) \cos (t)+(c_1 t+c_3) \sin (t) \\
\end{align*}
✓ Sympy. Time used: 0.183 (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")
x__4 = Function("x__4")
ode=[Eq(x__2(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + Derivative(x__2(t), t),0),Eq(-x__2(t) + x__4(t) + Derivative(x__3(t), t),0),Eq(-x__2(t) - x__3(t) + Derivative(x__4(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - \left (C_{1} - C_{2}\right ) \cos {\left (t \right )} + \left (C_{1} + C_{2}\right ) \sin {\left (t \right )}, \ x^{2}{\left (t \right )} = - \left (C_{1} - C_{2}\right ) \sin {\left (t \right )} - \left (C_{1} + C_{2}\right ) \cos {\left (t \right )}, \ x^{3}{\left (t \right )} = - C_{1} t \cos {\left (t \right )} + C_{2} t \sin {\left (t \right )} + \left (C_{1} - C_{4}\right ) \cos {\left (t \right )} - \left (C_{2} - C_{3}\right ) \sin {\left (t \right )}, \ x^{4}{\left (t \right )} = - C_{1} t \sin {\left (t \right )} - C_{2} t \cos {\left (t \right )} - C_{3} \cos {\left (t \right )} - C_{4} \sin {\left (t \right )}\right ]
\]