76.8.19 problem 19
Internal
problem
ID
[17437]
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
:
19
Date
solved
:
Monday, March 31, 2025 at 04:13:45 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=a x \left (t \right )+10 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )-4 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.116 (sec). Leaf size: 162
ode:=[diff(x(t),t) = a*x(t)+10*y(t), diff(y(t),t) = -x(t)-4*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (a -4+\sqrt {a^{2}+8 a -24}\right ) t}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (-a +4+\sqrt {a^{2}+8 a -24}\right ) t}{2}} \\
y \left (t \right ) &= \left (-\frac {a}{20}+\frac {\sqrt {a^{2}+8 a -24}}{20}-\frac {1}{5}\right ) c_1 \,{\mathrm e}^{\frac {\left (a -4+\sqrt {a^{2}+8 a -24}\right ) t}{2}}+\left (-\frac {{\mathrm e}^{-\frac {\left (-a +4+\sqrt {a^{2}+8 a -24}\right ) t}{2}} a}{20}-\frac {{\mathrm e}^{-\frac {\left (-a +4+\sqrt {a^{2}+8 a -24}\right ) t}{2}} \sqrt {a^{2}+8 a -24}}{20}-\frac {{\mathrm e}^{-\frac {\left (-a +4+\sqrt {a^{2}+8 a -24}\right ) t}{2}}}{5}\right ) c_2 \\
\end{align*}
✓ Mathematica. Time used: 0.009 (sec). Leaf size: 270
ode={D[x[t],t]==a*x[t]+10*y[t],D[y[t],t]==-x[t]-4*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {a^2+8 a-24}-a+4\right ) t} \left (c_1 \left (\left (\sqrt {a^2+8 a-24}+4\right ) e^{\sqrt {a^2+8 a-24} t}+a \left (e^{\sqrt {a^2+8 a-24} t}-1\right )+\sqrt {a^2+8 a-24}-4\right )+20 c_2 \left (e^{\sqrt {a^2+8 a-24} t}-1\right )\right )}{2 \sqrt {a^2+8 a-24}} \\
y(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {a^2+8 a-24}-a+4\right ) t} \left (c_2 \left (a \left (-e^{\sqrt {a^2+8 a-24} t}\right )+\left (\sqrt {a^2+8 a-24}-4\right ) e^{\sqrt {a^2+8 a-24} t}+\sqrt {a^2+8 a-24}+a+4\right )-2 c_1 \left (e^{\sqrt {a^2+8 a-24} t}-1\right )\right )}{2 \sqrt {a^2+8 a-24}} \\
\end{align*}
✓ Sympy. Time used: 0.235 (sec). Leaf size: 128
from sympy import *
t = symbols("t")
a = symbols("a")
x = Function("x")
y = Function("y")
ode=[Eq(-a*x(t) - 10*y(t) + Derivative(x(t), t),0),Eq(x(t) + 4*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} \left (a + \sqrt {a^{2} + 8 a - 24} + 4\right ) e^{\frac {t \left (a + \sqrt {a^{2} + 8 a - 24} - 4\right )}{2}}}{2} - \frac {C_{2} \left (a - \sqrt {a^{2} + 8 a - 24} + 4\right ) e^{- \frac {t \left (- a + \sqrt {a^{2} + 8 a - 24} + 4\right )}{2}}}{2}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (a + \sqrt {a^{2} + 8 a - 24} - 4\right )}{2}} + C_{2} e^{- \frac {t \left (- a + \sqrt {a^{2} + 8 a - 24} + 4\right )}{2}}\right ]
\]