64.24.7 problem 7
Internal
problem
ID
[13621]
Book
:
Differential
Equations
by
Shepley
L.
Ross.
Third
edition.
John
Willey.
New
Delhi.
2004.
Section
:
Chapter
13,
Nonlinear
differential
equations.
Section
13.2,
Exercises
page
656
Problem
number
:
7
Date
solved
:
Monday, March 31, 2025 at 08:02:57 AM
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 )+5 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.105 (sec). Leaf size: 81
ode:=[diff(x(t),t) = x(t)-y(t), diff(y(t),t) = x(t)+5*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{\left (3+\sqrt {3}\right ) t}+c_2 \,{\mathrm e}^{-\left (-3+\sqrt {3}\right ) t} \\
y \left (t \right ) &= -c_1 \,{\mathrm e}^{\left (3+\sqrt {3}\right ) t} \sqrt {3}+c_2 \,{\mathrm e}^{-\left (-3+\sqrt {3}\right ) t} \sqrt {3}-2 c_1 \,{\mathrm e}^{\left (3+\sqrt {3}\right ) t}-2 c_2 \,{\mathrm e}^{-\left (-3+\sqrt {3}\right ) t} \\
\end{align*}
✓ Mathematica. Time used: 0.008 (sec). Leaf size: 147
ode={D[x[t],t]==x[t]-y[t],D[y[t],t]==x[t]+5*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{6} e^{-\left (\left (\sqrt {3}-3\right ) t\right )} \left (c_1 \left (\left (3-2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+3+2 \sqrt {3}\right )-\sqrt {3} c_2 \left (e^{2 \sqrt {3} t}-1\right )\right ) \\
y(t)\to \frac {1}{6} e^{-\left (\left (\sqrt {3}-3\right ) t\right )} \left (\sqrt {3} c_1 \left (e^{2 \sqrt {3} t}-1\right )+c_2 \left (\left (3+2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+3-2 \sqrt {3}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.191 (sec). Leaf size: 66
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) - 5*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - C_{1} \left (\sqrt {3} + 2\right ) e^{t \left (3 - \sqrt {3}\right )} - C_{2} \left (2 - \sqrt {3}\right ) e^{t \left (\sqrt {3} + 3\right )}, \ y{\left (t \right )} = C_{1} e^{t \left (3 - \sqrt {3}\right )} + C_{2} e^{t \left (\sqrt {3} + 3\right )}\right ]
\]