82.56.4 problem Ex. 15
Internal
problem
ID
[18973]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
XI.
Ordinary
differential
equations
with
more
than
two
variables.
End
of
chapter
problems
at
page
143
Problem
number
:
Ex.
15
Date
solved
:
Monday, March 31, 2025 at 06:27:00 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=n y \left (t \right )-m z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=L z \left (t \right )-m x \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=m x \left (t \right )-L y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.676 (sec). Leaf size: 10311
ode:=[diff(x(t),t) = n*y(t)-m*z(t), diff(y(t),t) = L*z(t)-m*x(t), diff(z(t),t) = m*x(t)-L*y(t)];
dsolve(ode);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 0.013 (sec). Leaf size: 715
ode={D[x[t],t]==n*y[t]-m*z[t],D[y[t],t]==L*z[t]-m*x[t],D[z[t],t]==m*x[t]-L*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {L n e^{\text {$\#$1} t}-\text {$\#$1} m e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {L m e^{\text {$\#$1} t}+\text {$\#$1} n e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+L^2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ] \\
y(t)\to c_1 (-m) \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-L e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {\text {$\#$1} L e^{\text {$\#$1} t}+m^2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+m^2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ] \\
z(t)\to c_1 m \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {L e^{\text {$\#$1} t}+\text {$\#$1} e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {m n e^{\text {$\#$1} t}-\text {$\#$1} L e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+m n e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ] \\
\end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
L = symbols("L")
m = symbols("m")
n = symbols("n")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(m*z(t) - n*y(t) + Derivative(x(t), t),0),Eq(-L*z(t) + m*x(t) + Derivative(y(t), t),0),Eq(L*y(t) - m*x(t) + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
Timed Out