76.8.20 problem 20
Internal
problem
ID
[17430]
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
:
20
Date
solved
:
Thursday, March 13, 2025 at 10:08:14 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )+a y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=8 x \left (t \right )-6 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.095 (sec). Leaf size: 123
ode:=[diff(x(t),t) = 4*x(t)+a*y(t), diff(y(t),t) = 8*x(t)-6*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_{1} {\mathrm e}^{\left (-1+\sqrt {8 a +25}\right ) t}+c_{2} {\mathrm e}^{-t \sqrt {8 a +25}-t} \\
y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\left (-1+\sqrt {8 a +25}\right ) t} \sqrt {8 a +25}-c_{2} {\mathrm e}^{-t \sqrt {8 a +25}-t} \sqrt {8 a +25}-5 c_{1} {\mathrm e}^{\left (-1+\sqrt {8 a +25}\right ) t}-5 c_{2} {\mathrm e}^{-t \sqrt {8 a +25}-t}}{a} \\
\end{align*}
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 188
ode={D[x[t],t]==4*x[t]+a*y[t],D[y[t],t]==8*x[t]-6*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {e^{-\left (\left (\sqrt {8 a+25}+1\right ) t\right )} \left (c_1 \left (\left (\sqrt {8 a+25}+5\right ) e^{2 \sqrt {8 a+25} t}+\sqrt {8 a+25}-5\right )+a c_2 \left (e^{2 \sqrt {8 a+25} t}-1\right )\right )}{2 \sqrt {8 a+25}} \\
y(t)\to \frac {e^{-\left (\left (\sqrt {8 a+25}+1\right ) t\right )} \left (8 c_1 \left (e^{2 \sqrt {8 a+25} t}-1\right )+c_2 \left (\left (\sqrt {8 a+25}-5\right ) e^{2 \sqrt {8 a+25} t}+\sqrt {8 a+25}+5\right )\right )}{2 \sqrt {8 a+25}} \\
\end{align*}
✓ Sympy. Time used: 0.202 (sec). Leaf size: 88
from sympy import *
t = symbols("t")
a = symbols("a")
x = Function("x")
y = Function("y")
ode=[Eq(-a*y(t) - 4*x(t) + Derivative(x(t), t),0),Eq(-8*x(t) + 6*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} \left (\sqrt {8 a + 25} - 5\right ) e^{- t \left (\sqrt {8 a + 25} + 1\right )}}{8} + \frac {C_{2} \left (\sqrt {8 a + 25} + 5\right ) e^{t \left (\sqrt {8 a + 25} - 1\right )}}{8}, \ y{\left (t \right )} = C_{1} e^{- t \left (\sqrt {8 a + 25} + 1\right )} + C_{2} e^{t \left (\sqrt {8 a + 25} - 1\right )}\right ]
\]