14.29.3 problem 3
Internal
problem
ID
[2801]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
4.
Qualitative
theory
of
differential
equations.
Section
4.2
(Stability
of
linear
systems).
Page
383
Problem
number
:
3
Date
solved
:
Sunday, March 30, 2025 at 12:20:45 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-5 x \left (t \right )+3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.113 (sec). Leaf size: 80
ode:=[diff(x(t),t) = -5*x(t)+3*y(t), diff(y(t),t) = -x(t)+y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{\left (-2+\sqrt {6}\right ) t}+c_2 \,{\mathrm e}^{-\left (2+\sqrt {6}\right ) t} \\
y \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{\left (-2+\sqrt {6}\right ) t} \sqrt {6}}{3}-\frac {c_2 \,{\mathrm e}^{-\left (2+\sqrt {6}\right ) t} \sqrt {6}}{3}+c_1 \,{\mathrm e}^{\left (-2+\sqrt {6}\right ) t}+c_2 \,{\mathrm e}^{-\left (2+\sqrt {6}\right ) t} \\
\end{align*}
✓ Mathematica. Time used: 0.013 (sec). Leaf size: 143
ode={D[x[t],t]==-5*x[t]+3*y[t],D[y[t],t]==-1*x[t]+1*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{4} e^{-\left (\left (2+\sqrt {6}\right ) t\right )} \left (c_1 \left (-\left (\sqrt {6}-2\right ) e^{2 \sqrt {6} t}+2+\sqrt {6}\right )+\sqrt {6} c_2 \left (e^{2 \sqrt {6} t}-1\right )\right ) \\
y(t)\to \frac {1}{12} e^{-\left (\left (2+\sqrt {6}\right ) t\right )} \left (3 c_2 \left (\left (2+\sqrt {6}\right ) e^{2 \sqrt {6} t}+2-\sqrt {6}\right )-\sqrt {6} c_1 \left (e^{2 \sqrt {6} t}-1\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.191 (sec). Leaf size: 65
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(5*x(t) - 3*y(t) + Derivative(x(t), t),0),Eq(x(t) - y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{1} \left (3 - \sqrt {6}\right ) e^{- t \left (2 - \sqrt {6}\right )} + C_{2} \left (\sqrt {6} + 3\right ) e^{- t \left (2 + \sqrt {6}\right )}, \ y{\left (t \right )} = C_{1} e^{- t \left (2 - \sqrt {6}\right )} + C_{2} e^{- t \left (2 + \sqrt {6}\right )}\right ]
\]