76.8.16 problem 16
Internal
problem
ID
[17434]
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
:
16
Date
solved
:
Monday, March 31, 2025 at 04:13:41 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=\frac {5 x \left (t \right )}{4}+\frac {3 y \left (t \right )}{4}\\ \frac {d}{d t}y \left (t \right )&=a x \left (t \right )+\frac {5 y \left (t \right )}{4} \end{align*}
✓ Maple. Time used: 0.124 (sec). Leaf size: 82
ode:=[diff(x(t),t) = 5/4*x(t)+3/4*y(t), diff(y(t),t) = a*x(t)+5/4*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (5+2 \sqrt {3}\, \sqrt {a}\right ) t}{4}}+c_2 \,{\mathrm e}^{-\frac {\left (2 \sqrt {3}\, \sqrt {a}-5\right ) t}{4}} \\
y \left (t \right ) &= \frac {2 \sqrt {3}\, \sqrt {a}\, \left (c_1 \,{\mathrm e}^{\frac {\left (5+2 \sqrt {3}\, \sqrt {a}\right ) t}{4}}-c_2 \,{\mathrm e}^{-\frac {\left (2 \sqrt {3}\, \sqrt {a}-5\right ) t}{4}}\right )}{3} \\
\end{align*}
✓ Mathematica. Time used: 0.006 (sec). Leaf size: 164
ode={D[x[t],t]==5/4*x[t]+3/4*y[t],D[y[t],t]==a*x[t]+5/4*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {e^{\frac {1}{4} \left (5-2 \sqrt {3} \sqrt {a}\right ) t} \left (2 \sqrt {a} c_1 \left (e^{\sqrt {3} \sqrt {a} t}+1\right )+\sqrt {3} c_2 \left (e^{\sqrt {3} \sqrt {a} t}-1\right )\right )}{4 \sqrt {a}} \\
y(t)\to \frac {1}{6} e^{\frac {1}{4} \left (5-2 \sqrt {3} \sqrt {a}\right ) t} \left (2 \sqrt {3} \sqrt {a} c_1 \left (e^{\sqrt {3} \sqrt {a} t}-1\right )+3 c_2 \left (e^{\sqrt {3} \sqrt {a} t}+1\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.207 (sec). Leaf size: 109
from sympy import *
t = symbols("t")
a = symbols("a")
x = Function("x")
y = Function("y")
ode=[Eq(-5*x(t)/4 - 3*y(t)/4 + Derivative(x(t), t),0),Eq(-a*x(t) - 5*y(t)/4 + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {\sqrt {3} C_{1} e^{- \frac {t \left (2 \sqrt {3} \sqrt {a} - 5\right )}{4}}}{2 \sqrt {a}} + \frac {\sqrt {3} C_{2} e^{\frac {t \left (2 \sqrt {3} \sqrt {a} + 5\right )}{4}}}{2 \sqrt {a}}, \ y{\left (t \right )} = C_{1} e^{- \frac {t \left (2 \sqrt {3} \sqrt {a} - 5\right )}{4}} + C_{2} e^{\frac {t \left (2 \sqrt {3} \sqrt {a} + 5\right )}{4}}\right ]
\]