70.1.3 problem 2.1 (iii)
Internal
problem
ID
[14214]
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
(iii)
Date
solved
:
Wednesday, March 05, 2025 at 10:40:14 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-4 x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )-2 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.036 (sec). Leaf size: 82
ode:=[diff(x(t),t) = -4*x(t)+2*y(t), diff(y(t),t) = 3*x(t)-2*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_{1} {\mathrm e}^{\left (-3+\sqrt {7}\right ) t}+c_{2} {\mathrm e}^{-\left (3+\sqrt {7}\right ) t} \\
y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\left (-3+\sqrt {7}\right ) t} \sqrt {7}}{2}-\frac {c_{2} {\mathrm e}^{-\left (3+\sqrt {7}\right ) t} \sqrt {7}}{2}+\frac {c_{1} {\mathrm e}^{\left (-3+\sqrt {7}\right ) t}}{2}+\frac {c_{2} {\mathrm e}^{-\left (3+\sqrt {7}\right ) t}}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.008 (sec). Leaf size: 143
ode={D[x[t],t]==-4*x[t]+2*y[t],D[y[t],t]==3*x[t]-2*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{14} e^{-\left (\left (3+\sqrt {7}\right ) t\right )} \left (c_1 \left (-\left (\sqrt {7}-7\right ) e^{2 \sqrt {7} t}+7+\sqrt {7}\right )+2 \sqrt {7} c_2 \left (e^{2 \sqrt {7} t}-1\right )\right ) \\
y(t)\to \frac {1}{14} e^{-\left (\left (3+\sqrt {7}\right ) t\right )} \left (3 \sqrt {7} c_1 \left (e^{2 \sqrt {7} t}-1\right )+c_2 \left (\left (7+\sqrt {7}\right ) e^{2 \sqrt {7} t}+7-\sqrt {7}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.208 (sec). Leaf size: 70
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(4*x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(-3*x(t) + 2*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} \left (1 - \sqrt {7}\right ) e^{- t \left (3 - \sqrt {7}\right )}}{3} - \frac {C_{2} \left (1 + \sqrt {7}\right ) e^{- t \left (\sqrt {7} + 3\right )}}{3}, \ y{\left (t \right )} = C_{1} e^{- t \left (3 - \sqrt {7}\right )} + C_{2} e^{- t \left (\sqrt {7} + 3\right )}\right ]
\]