14.30.3 problem 3
Internal
problem
ID
[2816]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
4.
Qualitative
theory
of
differential
equations.
Section
4.3
(Stability
of
equilibrium
solutions).
Page
393
Problem
number
:
3
Date
solved
:
Tuesday, January 28, 2025 at 02:38:54 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }&=x^{2}+y \left (t \right )^{2}-1\\ y^{\prime }\left (t \right )&=2 x y \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 1.504 (sec). Leaf size: 257
dsolve([diff(x(t),t)=x(t)^2+y(t)^2-1,diff(y(t),t)=2*x(t)*y(t)],singsol=all)
\begin{align*}
\\
\left [\left \{y &= \frac {c_1 \,{\mathrm e}^{-2 t}}{4}, y = \frac {c_1 \,{\mathrm e}^{-2 t} \left (-\frac {8 c_2 \,c_1^{2} \left ({\mathrm e}^{-2 t}+2 c_2 \right )}{c_1^{2} {\mathrm e}^{-4 t}+4 \,{\mathrm e}^{-2 t} c_1^{2} c_2 +4 c_1^{2} c_2^{2}-64}-\frac {4 \,{\mathrm e}^{-2 t} c_1^{2} \left ({\mathrm e}^{-2 t}+2 c_2 \right )}{c_1^{2} {\mathrm e}^{-4 t}+4 \,{\mathrm e}^{-2 t} c_1^{2} c_2 +4 c_1^{2} c_2^{2}-64}+4\right )}{16}, y = -\frac {c_1 \,{\mathrm e}^{-2 t} \left (\frac {8 c_2 \,c_1^{2} \left ({\mathrm e}^{-2 t}+2 c_2 \right )}{c_1^{2} {\mathrm e}^{-4 t}+4 \,{\mathrm e}^{-2 t} c_1^{2} c_2 +4 c_1^{2} c_2^{2}-64}+\frac {4 \,{\mathrm e}^{-2 t} c_1^{2} \left ({\mathrm e}^{-2 t}+2 c_2 \right )}{c_1^{2} {\mathrm e}^{-4 t}+4 \,{\mathrm e}^{-2 t} c_1^{2} c_2 +4 c_1^{2} c_2^{2}-64}-4\right )}{16}\right \}, \left \{x \left (t \right ) = \frac {y^{\prime }}{2 y}\right \}\right ] \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.254 (sec). Leaf size: 1239
DSolve[{D[x[t],t]==x[t]^2+y[t]^2-1,D[y[t],t]==2*x[t]*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
y(t)\to \frac {1}{2} \left (c_1-\sqrt {\frac {\left (\left (-4+c_1{}^2\right ) \cosh (8 (t+2 c_2))+\left (-4+c_1{}^2\right ) \sinh (8 (t+2 c_2))-4 \sqrt {c_1{}^2 \left (-4+c_1{}^2\right ) \sinh ^2(2 (t+2 c_2)) (\cosh (8 (t+2 c_2))+\sinh (8 (t+2 c_2)))}+4-c_1{}^2\right ){}^2}{\left (-\left (-4+c_1{}^2\right ) \cosh (4 (t+2 c_2))+4+c_1{}^2\right ){}^2 (\cosh (4 (t+2 c_2))+\sinh (4 (t+2 c_2))){}^2}-4+c_1{}^2}\right ) \\
x(t)\to \frac {2 \cosh (4 (t+2 c_2)) \sqrt {c_1{}^2 \left (-4+c_1{}^2\right ) \sinh ^2(2 (t+2 c_2)) (\cosh (8 (t+2 c_2))+\sinh (8 (t+2 c_2)))}-\sinh (4 (t+2 c_2)) \left (2 \sqrt {c_1{}^2 \left (-4+c_1{}^2\right ) \sinh ^2(2 (t+2 c_2)) (\cosh (8 (t+2 c_2))+\sinh (8 (t+2 c_2)))}-4+c_1{}^2\right )}{\left (-4+c_1{}^2\right ) \cosh (4 (t+2 c_2))-4-c_1{}^2} \\
y(t)\to \frac {1}{2} \left (\sqrt {\frac {\left (\left (-4+c_1{}^2\right ) \cosh (8 (t+2 c_2))+\left (-4+c_1{}^2\right ) \sinh (8 (t+2 c_2))-4 \sqrt {c_1{}^2 \left (-4+c_1{}^2\right ) \sinh ^2(2 (t+2 c_2)) (\cosh (8 (t+2 c_2))+\sinh (8 (t+2 c_2)))}+4-c_1{}^2\right ){}^2}{\left (-\left (-4+c_1{}^2\right ) \cosh (4 (t+2 c_2))+4+c_1{}^2\right ){}^2 (\cosh (4 (t+2 c_2))+\sinh (4 (t+2 c_2))){}^2}-4+c_1{}^2}+c_1\right ) \\
x(t)\to \frac {2 \cosh (4 (t+2 c_2)) \sqrt {c_1{}^2 \left (-4+c_1{}^2\right ) \sinh ^2(2 (t+2 c_2)) (\cosh (8 (t+2 c_2))+\sinh (8 (t+2 c_2)))}-\sinh (4 (t+2 c_2)) \left (2 \sqrt {c_1{}^2 \left (-4+c_1{}^2\right ) \sinh ^2(2 (t+2 c_2)) (\cosh (8 (t+2 c_2))+\sinh (8 (t+2 c_2)))}-4+c_1{}^2\right )}{\left (-4+c_1{}^2\right ) \cosh (4 (t+2 c_2))-4-c_1{}^2} \\
y(t)\to \frac {1}{2} \left (c_1-\sqrt {\frac {\left (\left (-4+c_1{}^2\right ) \cosh (8 (t+2 c_2))+\left (-4+c_1{}^2\right ) \sinh (8 (t+2 c_2))+4 \sqrt {c_1{}^2 \left (-4+c_1{}^2\right ) \sinh ^2(2 (t+2 c_2)) (\cosh (8 (t+2 c_2))+\sinh (8 (t+2 c_2)))}+4-c_1{}^2\right ){}^2}{\left (-\left (-4+c_1{}^2\right ) \cosh (4 (t+2 c_2))+4+c_1{}^2\right ){}^2 (\cosh (4 (t+2 c_2))+\sinh (4 (t+2 c_2))){}^2}-4+c_1{}^2}\right ) \\
x(t)\to \frac {\sinh (4 (t+2 c_2)) \left (2 \sqrt {c_1{}^2 \left (-4+c_1{}^2\right ) \sinh ^2(2 (t+2 c_2)) (\cosh (8 (t+2 c_2))+\sinh (8 (t+2 c_2)))}+4-c_1{}^2\right )-2 \cosh (4 (t+2 c_2)) \sqrt {c_1{}^2 \left (-4+c_1{}^2\right ) \sinh ^2(2 (t+2 c_2)) (\cosh (8 (t+2 c_2))+\sinh (8 (t+2 c_2)))}}{\left (-4+c_1{}^2\right ) \cosh (4 (t+2 c_2))-4-c_1{}^2} \\
y(t)\to \frac {1}{2} \left (\sqrt {\frac {\left (\left (-4+c_1{}^2\right ) \cosh (8 (t+2 c_2))+\left (-4+c_1{}^2\right ) \sinh (8 (t+2 c_2))+4 \sqrt {c_1{}^2 \left (-4+c_1{}^2\right ) \sinh ^2(2 (t+2 c_2)) (\cosh (8 (t+2 c_2))+\sinh (8 (t+2 c_2)))}+4-c_1{}^2\right ){}^2}{\left (-\left (-4+c_1{}^2\right ) \cosh (4 (t+2 c_2))+4+c_1{}^2\right ){}^2 (\cosh (4 (t+2 c_2))+\sinh (4 (t+2 c_2))){}^2}-4+c_1{}^2}+c_1\right ) \\
x(t)\to \frac {\sinh (4 (t+2 c_2)) \left (2 \sqrt {c_1{}^2 \left (-4+c_1{}^2\right ) \sinh ^2(2 (t+2 c_2)) (\cosh (8 (t+2 c_2))+\sinh (8 (t+2 c_2)))}+4-c_1{}^2\right )-2 \cosh (4 (t+2 c_2)) \sqrt {c_1{}^2 \left (-4+c_1{}^2\right ) \sinh ^2(2 (t+2 c_2)) (\cosh (8 (t+2 c_2))+\sinh (8 (t+2 c_2)))}}{\left (-4+c_1{}^2\right ) \cosh (4 (t+2 c_2))-4-c_1{}^2} \\
\end{align*}