70.1.6 problem 2.1 (vi)
Internal
problem
ID
[14217]
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
(vi)
Date
solved
:
Wednesday, March 05, 2025 at 10:40:18 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )+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: 0.046 (sec). Leaf size: 81
ode:=[diff(x(t),t) = 2*x(t)+y(t), diff(y(t),t) = y(t)-x(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \left (\sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} \right ) \\
y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {3 t}{2}} \left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} -\sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} \right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.016 (sec). Leaf size: 111
ode={D[x[t],t]==2*x[t]+y[t],D[y[t],t]==-x[t]+y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{3} e^{3 t/2} \left (3 c_1 \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} (c_1+2 c_2) \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \\
y(t)\to \frac {1}{3} e^{3 t/2} \left (3 c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )-\sqrt {3} (2 c_1+c_2) \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.206 (sec). Leaf size: 99
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-2*x(t) - 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)
\[
\left [ x{\left (t \right )} = - \left (\frac {C_{1}}{2} - \frac {\sqrt {3} C_{2}}{2}\right ) e^{\frac {3 t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} + \left (\frac {\sqrt {3} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{\frac {3 t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}, \ y{\left (t \right )} = C_{1} e^{\frac {3 t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} - C_{2} e^{\frac {3 t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}\right ]
\]