70.1.2 problem 2.1 (ii)
Internal
problem
ID
[14213]
Book
:
Nonlinear
Ordinary
Differential
Equations
by
D.W.Jordna
and
P.Smith.
4th
edition
1999.
Oxford
Univ.
Press.
NY
Section
:
Chapter
2.
Plane
autonomous
systems
and
linearization.
Problems
page
79
Problem
number
:
2.1
(ii)
Date
solved
:
Wednesday, March 05, 2025 at 10:40:13 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-2 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.039 (sec). Leaf size: 85
ode:=[diff(x(t),t) = x(t)+y(t), diff(y(t),t) = x(t)-2*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {13}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {13}\right ) t}{2}} \\
y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {13}\right ) t}{2}} \sqrt {13}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {13}\right ) t}{2}} \sqrt {13}}{2}-\frac {3 c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {13}\right ) t}{2}}}{2}-\frac {3 c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {13}\right ) t}{2}}}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.009 (sec). Leaf size: 149
ode={D[x[t],t]==x[t]+y[t],D[y[t],t]==x[t]-2*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{26} e^{-\frac {1}{2} \left (1+\sqrt {13}\right ) t} \left (c_1 \left (\left (13+3 \sqrt {13}\right ) e^{\sqrt {13} t}+13-3 \sqrt {13}\right )+2 \sqrt {13} c_2 \left (e^{\sqrt {13} t}-1\right )\right ) \\
y(t)\to \frac {1}{26} e^{-\frac {1}{2} \left (1+\sqrt {13}\right ) t} \left (2 \sqrt {13} c_1 \left (e^{\sqrt {13} t}-1\right )-c_2 \left (\left (3 \sqrt {13}-13\right ) e^{\sqrt {13} t}-13-3 \sqrt {13}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.191 (sec). Leaf size: 75
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t) - y(t) + Derivative(x(t), t),0),Eq(-x(t) + 2*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {C_{1} \left (3 - \sqrt {13}\right ) e^{- \frac {t \left (1 + \sqrt {13}\right )}{2}}}{2} + \frac {C_{2} \left (3 + \sqrt {13}\right ) e^{- \frac {t \left (1 - \sqrt {13}\right )}{2}}}{2}, \ y{\left (t \right )} = C_{1} e^{- \frac {t \left (1 + \sqrt {13}\right )}{2}} + C_{2} e^{- \frac {t \left (1 - \sqrt {13}\right )}{2}}\right ]
\]