76.8.18 problem 18
Internal
problem
ID
[17436]
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
:
18
Date
solved
:
Monday, March 31, 2025 at 04:13:44 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )+a y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-6 x \left (t \right )-4 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.124 (sec). Leaf size: 121
ode:=[diff(x(t),t) = 3*x(t)+a*y(t), diff(y(t),t) = -6*x(t)-4*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (-1+\sqrt {49-24 a}\right ) t}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (1+\sqrt {49-24 a}\right ) t}{2}} \\
y \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{\frac {\left (-1+\sqrt {49-24 a}\right ) t}{2}} \sqrt {49-24 a}-c_2 \,{\mathrm e}^{-\frac {\left (1+\sqrt {49-24 a}\right ) t}{2}} \sqrt {49-24 a}-7 c_1 \,{\mathrm e}^{\frac {\left (-1+\sqrt {49-24 a}\right ) t}{2}}-7 c_2 \,{\mathrm e}^{-\frac {\left (1+\sqrt {49-24 a}\right ) t}{2}}}{2 a} \\
\end{align*}
✓ Mathematica. Time used: 0.008 (sec). Leaf size: 189
ode={D[x[t],t]==3*x[t]+a*y[t],D[y[t],t]==-6*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 {49-24 a}+1\right ) t} \left (c_1 \left (\left (\sqrt {49-24 a}+7\right ) e^{\sqrt {49-24 a} t}+\sqrt {49-24 a}-7\right )+2 a c_2 \left (e^{\sqrt {49-24 a} t}-1\right )\right )}{2 \sqrt {49-24 a}} \\
y(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {49-24 a}+1\right ) t} \left (c_2 \left (\left (\sqrt {49-24 a}-7\right ) e^{\sqrt {49-24 a} t}+\sqrt {49-24 a}+7\right )-12 c_1 \left (e^{\sqrt {49-24 a} t}-1\right )\right )}{2 \sqrt {49-24 a}} \\
\end{align*}
✓ Sympy. Time used: 0.203 (sec). Leaf size: 95
from sympy import *
t = symbols("t")
a = symbols("a")
x = Function("x")
y = Function("y")
ode=[Eq(-a*y(t) - 3*x(t) + Derivative(x(t), t),0),Eq(6*x(t) + 4*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} \left (\sqrt {49 - 24 a} + 7\right ) e^{\frac {t \left (\sqrt {49 - 24 a} - 1\right )}{2}}}{12} + \frac {C_{2} \left (\sqrt {49 - 24 a} - 7\right ) e^{- \frac {t \left (\sqrt {49 - 24 a} + 1\right )}{2}}}{12}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (\sqrt {49 - 24 a} - 1\right )}{2}} + C_{2} e^{- \frac {t \left (\sqrt {49 - 24 a} + 1\right )}{2}}\right ]
\]