76.8.19 problem 19
Internal
problem
ID
[17508]
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
:
Tuesday, January 28, 2025 at 10:39:42 AM
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*}
✓ Solution by Maple
Time used: 0.111 (sec). Leaf size: 162
dsolve([diff(x(t),t)=a*x(t)+10*y(t),diff(y(t),t)=-x(t)-4*y(t)],singsol=all)
\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*}
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 270
DSolve[{D[x[t],t]==a*x[t]+10*y[t],D[y[t],t]==-x[t]-4*y[t]},{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*}