70.1.4 problem 2.1 (iv)
Internal
problem
ID
[14215]
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
(iv)
Date
solved
:
Wednesday, March 05, 2025 at 10:40:16 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+2 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.037 (sec). Leaf size: 85
ode:=[diff(x(t),t) = x(t)+2*y(t), diff(y(t),t) = 2*x(t)+2*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (3+\sqrt {17}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (-3+\sqrt {17}\right ) t}{2}} \\
y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\frac {\left (3+\sqrt {17}\right ) t}{2}} \sqrt {17}}{4}-\frac {c_{2} {\mathrm e}^{-\frac {\left (-3+\sqrt {17}\right ) t}{2}} \sqrt {17}}{4}+\frac {c_{1} {\mathrm e}^{\frac {\left (3+\sqrt {17}\right ) t}{2}}}{4}+\frac {c_{2} {\mathrm e}^{-\frac {\left (-3+\sqrt {17}\right ) t}{2}}}{4} \\
\end{align*}
✓ Mathematica. Time used: 0.009 (sec). Leaf size: 143
ode={D[x[t],t]==x[t]+2*y[t],D[y[t],t]==2*x[t]+2*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{34} e^{-\frac {1}{2} \left (\sqrt {17}-3\right ) t} \left (c_1 \left (-\left (\sqrt {17}-17\right ) e^{\sqrt {17} t}+17+\sqrt {17}\right )+4 \sqrt {17} c_2 \left (e^{\sqrt {17} t}-1\right )\right ) \\
y(t)\to \frac {1}{34} e^{-\frac {1}{2} \left (\sqrt {17}-3\right ) t} \left (4 \sqrt {17} c_1 \left (e^{\sqrt {17} t}-1\right )+c_2 \left (\left (17+\sqrt {17}\right ) e^{\sqrt {17} t}+17-\sqrt {17}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.199 (sec). Leaf size: 76
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(-2*x(t) - 2*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} \left (1 - \sqrt {17}\right ) e^{\frac {t \left (3 + \sqrt {17}\right )}{2}}}{4} - \frac {C_{2} \left (1 + \sqrt {17}\right ) e^{\frac {t \left (3 - \sqrt {17}\right )}{2}}}{4}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (3 + \sqrt {17}\right )}{2}} + C_{2} e^{\frac {t \left (3 - \sqrt {17}\right )}{2}}\right ]
\]