72.14.1 problem 1
Internal
problem
ID
[14785]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
3.
Linear
Systems.
Exercises
section
3.8
page
371
Problem
number
:
1
Date
solved
:
Thursday, March 13, 2025 at 04:18:56 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=\frac {y \left (t \right )}{10}\\ \frac {d}{d t}y \left (t \right )&=\frac {z \left (t \right )}{5}\\ \frac {d}{d t}z \left (t \right )&=\frac {2 x \left (t \right )}{5} \end{align*}
✓ Maple. Time used: 0.085 (sec). Leaf size: 182
ode:=[diff(x(t),t) = 1/10*y(t), diff(y(t),t) = 1/5*z(t), diff(z(t),t) = 2/5*x(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \frac {{\mathrm e}^{\frac {t}{5}} c_{1}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {t}{10}} \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{4}+\frac {c_{2} {\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{4}-\frac {c_{3} {\mathrm e}^{-\frac {t}{10}} \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{4}-\frac {c_{3} {\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{4} \\
y &= {\mathrm e}^{\frac {t}{5}} c_{1} -\frac {c_{2} {\mathrm e}^{-\frac {t}{10}} \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{2}-\frac {c_{2} {\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{2}-\frac {c_{3} {\mathrm e}^{-\frac {t}{10}} \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{2}+\frac {c_{3} {\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{2} \\
z &= {\mathrm e}^{\frac {t}{5}} c_{1} +c_{2} {\mathrm e}^{-\frac {t}{10}} \sin \left (\frac {\sqrt {3}\, t}{10}\right )+c_{3} {\mathrm e}^{-\frac {t}{10}} \cos \left (\frac {\sqrt {3}\, t}{10}\right ) \\
\end{align*}
✓ Mathematica. Time used: 0.03 (sec). Leaf size: 269
ode={D[x[t],t]==0*x[t]+1/10*y[t]+0*z[t],D[y[t],t]==0*x[t]+0*y[t]+2/10*z[t],D[z[t],t]==4/10*x[t]+0*y[t]+0*z[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{6} e^{-t/10} \left ((2 c_1+c_2+c_3) e^{t/10} \sqrt [5]{e^t}+(4 c_1-c_2-c_3) \cos \left (\frac {\sqrt {3} t}{10}\right )+\sqrt {3} (c_2-c_3) \sin \left (\frac {\sqrt {3} t}{10}\right )\right ) \\
y(t)\to \frac {1}{3} e^{-t/10} \left ((2 c_1+c_2+c_3) e^{t/10} \sqrt [5]{e^t}-(2 c_1-2 c_2+c_3) \cos \left (\frac {\sqrt {3} t}{10}\right )-\sqrt {3} (2 c_1-c_3) \sin \left (\frac {\sqrt {3} t}{10}\right )\right ) \\
z(t)\to \frac {1}{3} e^{-t/10} \left ((2 c_1+c_2+c_3) e^{t/10} \sqrt [5]{e^t}-(2 c_1+c_2-2 c_3) \cos \left (\frac {\sqrt {3} t}{10}\right )+\sqrt {3} (2 c_1-c_2) \sin \left (\frac {\sqrt {3} t}{10}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.302 (sec). Leaf size: 170
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(-y(t)/10 + Derivative(x(t), t),0),Eq(-z(t)/5 + Derivative(y(t), t),0),Eq(-2*x(t)/5 + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {C_{3} e^{\frac {t}{5}}}{2} - \left (\frac {C_{1}}{4} + \frac {\sqrt {3} C_{2}}{4}\right ) e^{- \frac {t}{10}} \cos {\left (\frac {\sqrt {3} t}{10} \right )} - \left (\frac {\sqrt {3} C_{1}}{4} - \frac {C_{2}}{4}\right ) e^{- \frac {t}{10}} \sin {\left (\frac {\sqrt {3} t}{10} \right )}, \ y{\left (t \right )} = C_{3} e^{\frac {t}{5}} - \left (\frac {C_{1}}{2} - \frac {\sqrt {3} C_{2}}{2}\right ) e^{- \frac {t}{10}} \cos {\left (\frac {\sqrt {3} t}{10} \right )} + \left (\frac {\sqrt {3} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- \frac {t}{10}} \sin {\left (\frac {\sqrt {3} t}{10} \right )}, \ z{\left (t \right )} = C_{1} e^{- \frac {t}{10}} \cos {\left (\frac {\sqrt {3} t}{10} \right )} - C_{2} e^{- \frac {t}{10}} \sin {\left (\frac {\sqrt {3} t}{10} \right )} + C_{3} e^{\frac {t}{5}}\right ]
\]