67.5.15 problem Problem 3(d)
Internal
problem
ID
[14030]
Book
:
APPLIED
DIFFERENTIAL
EQUATIONS
The
Primary
Course
by
Vladimir
A.
Dobrushkin.
CRC
Press
2015
Section
:
Chapter
6.
Introduction
to
Systems
of
ODEs.
Problems
page
408
Problem
number
:
Problem
3(d)
Date
solved
:
Monday, March 31, 2025 at 08:22:27 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )+5 x \left (t \right )-2 y \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )+2 x \left (t \right )-y \left (t \right )&=0 \end{align*}
✓ Maple. Time used: 0.107 (sec). Leaf size: 82
ode:=[diff(x(t),t)+5*x(t)-2*y(t) = 0, diff(y(t),t)+2*x(t)-y(t) = 0];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{\left (-2+\sqrt {5}\right ) t}+c_2 \,{\mathrm e}^{-\left (2+\sqrt {5}\right ) t} \\
y \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{\left (-2+\sqrt {5}\right ) t} \sqrt {5}}{2}-\frac {c_2 \,{\mathrm e}^{-\left (2+\sqrt {5}\right ) t} \sqrt {5}}{2}+\frac {3 c_1 \,{\mathrm e}^{\left (-2+\sqrt {5}\right ) t}}{2}+\frac {3 c_2 \,{\mathrm e}^{-\left (2+\sqrt {5}\right ) t}}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.008 (sec). Leaf size: 148
ode={D[x[t],t]+5*x[t]-2*y[t]==0,D[y[t],t]+2*x[t]-y[t]==0};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{10} e^{-\left (\left (2+\sqrt {5}\right ) t\right )} \left (c_1 \left (\left (5-3 \sqrt {5}\right ) e^{2 \sqrt {5} t}+5+3 \sqrt {5}\right )+2 \sqrt {5} c_2 \left (e^{2 \sqrt {5} t}-1\right )\right ) \\
y(t)\to \frac {1}{10} e^{-\left (\left (2+\sqrt {5}\right ) t\right )} \left (c_2 \left (\left (5+3 \sqrt {5}\right ) e^{2 \sqrt {5} t}+5-3 \sqrt {5}\right )-2 \sqrt {5} c_1 \left (e^{2 \sqrt {5} t}-1\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.213 (sec). Leaf size: 68
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(5*x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(2*x(t) - y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {C_{1} \left (3 - \sqrt {5}\right ) e^{- t \left (2 - \sqrt {5}\right )}}{2} + \frac {C_{2} \left (\sqrt {5} + 3\right ) e^{- t \left (2 + \sqrt {5}\right )}}{2}, \ y{\left (t \right )} = C_{1} e^{- t \left (2 - \sqrt {5}\right )} + C_{2} e^{- t \left (2 + \sqrt {5}\right )}\right ]
\]