76.8.14 problem 14
Internal
problem
ID
[17432]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
3.
Systems
of
two
first
order
equations.
Section
3.4
(Complex
Eigenvalues).
Problems
at
page
177
Problem
number
:
14
Date
solved
:
Monday, March 31, 2025 at 04:13:37 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+a y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.112 (sec). Leaf size: 95
ode:=[diff(x(t),t) = -5*y(t), diff(y(t),t) = x(t)+a*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (a +\sqrt {a^{2}-20}\right ) t}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (-a +\sqrt {a^{2}-20}\right ) t}{2}} \\
y \left (t \right ) &= -\frac {c_1 \left (a +\sqrt {a^{2}-20}\right ) {\mathrm e}^{\frac {\left (a +\sqrt {a^{2}-20}\right ) t}{2}}}{10}-\frac {c_2 \left (a -\sqrt {a^{2}-20}\right ) {\mathrm e}^{\frac {\left (a -\sqrt {a^{2}-20}\right ) t}{2}}}{10} \\
\end{align*}
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 207
ode={D[x[t],t]==-5*y[t],D[y[t],t]==x[t]+a*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {e^{\frac {1}{2} \left (a-\sqrt {a^2-20}\right ) t} \left (-a c_1 \left (e^{\sqrt {a^2-20} t}-1\right )+\sqrt {a^2-20} c_1 \left (e^{\sqrt {a^2-20} t}+1\right )-10 c_2 \left (e^{\sqrt {a^2-20} t}-1\right )\right )}{2 \sqrt {a^2-20}} \\
y(t)\to \frac {e^{\frac {1}{2} \left (a-\sqrt {a^2-20}\right ) t} \left (2 c_1 \left (e^{\sqrt {a^2-20} t}-1\right )+c_2 \left (a \left (e^{\sqrt {a^2-20} t}-1\right )+\sqrt {a^2-20} \left (e^{\sqrt {a^2-20} t}+1\right )\right )\right )}{2 \sqrt {a^2-20}} \\
\end{align*}
✓ Sympy. Time used: 0.195 (sec). Leaf size: 97
from sympy import *
t = symbols("t")
a = symbols("a")
x = Function("x")
y = Function("y")
ode=[Eq(5*y(t) + Derivative(x(t), t),0),Eq(-a*y(t) - x(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} \left (a + \sqrt {a^{2} - 20}\right ) e^{\frac {t \left (a - \sqrt {a^{2} - 20}\right )}{2}}}{2} - \frac {C_{2} \left (a - \sqrt {a^{2} - 20}\right ) e^{\frac {t \left (a + \sqrt {a^{2} - 20}\right )}{2}}}{2}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (a - \sqrt {a^{2} - 20}\right )}{2}} + C_{2} e^{\frac {t \left (a + \sqrt {a^{2} - 20}\right )}{2}}\right ]
\]