75.26.7 problem 774
Internal
problem
ID
[17235]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Chapter
3
(Systems
of
differential
equations).
Section
19.
Basic
concepts
and
definitions.
Exercises
page
199
Problem
number
:
774
Date
solved
:
Tuesday, January 28, 2025 at 08:27:35 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=\frac {t -y \left (t \right )}{y \left (t \right )-x \left (t \right )}\\ \frac {d}{d t}y \left (t \right )&=\frac {x \left (t \right )-t}{y \left (t \right )-x \left (t \right )} \end{align*}
✓ Solution by Maple
Time used: 3.527 (sec). Leaf size: 131
dsolve([diff(x(t),t)=(t-y(t))/(y(t)-x(t)),diff(y(t),t)=(x(t)-t)/(y(t)-x(t))],singsol=all)
\begin{align*}
\left \{x \left (t \right ) &= t +\operatorname {RootOf}\left (-t +\int _{}^{\textit {\_Z}}-\frac {2 \left ({\mathrm e}^{c_{1}} \textit {\_f}^{2}-1\right )}{3 \,{\mathrm e}^{c_{1}} \textit {\_f}^{2}-{\mathrm e}^{\frac {c_{1}}{2}} \sqrt {-3 \,{\mathrm e}^{c_{1}} \textit {\_f}^{2}+4}\, \textit {\_f} -4}d \textit {\_f} +c_{2} \right ), x \left (t \right ) = t +\operatorname {RootOf}\left (-t +\int _{}^{\textit {\_Z}}-\frac {2 \left ({\mathrm e}^{c_{1}} \textit {\_f}^{2}-1\right )}{3 \,{\mathrm e}^{c_{1}} \textit {\_f}^{2}+{\mathrm e}^{\frac {c_{1}}{2}} \sqrt {-3 \,{\mathrm e}^{c_{1}} \textit {\_f}^{2}+4}\, \textit {\_f} -4}d \textit {\_f} +c_{2} \right )\right \} \\
\left \{y \left (t \right ) &= \frac {x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )+t}{\frac {d}{d t}x \left (t \right )+1}\right \} \\
\end{align*}
✓ Solution by Mathematica
Time used: 5.742 (sec). Leaf size: 151
DSolve[{D[x[t],t]==(t-y[t])/(y[t]-x[t]),D[y[t],t]==(x[t]-t)/(y[t]-x[t])},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to \frac {1}{2} \left (-\sqrt {-3 t^2+2 c_1 t+c_1{}^2+4 c_2}-t+c_1\right ) \\
y(t)\to \frac {1}{2} \left (\sqrt {-3 t^2+2 c_1 t+c_1{}^2+4 c_2}-t+c_1\right ) \\
x(t)\to \frac {1}{2} \left (\sqrt {-3 t^2+2 c_1 t+c_1{}^2+4 c_2}-t+c_1\right ) \\
y(t)\to \frac {1}{2} \left (-\sqrt {-3 t^2+2 c_1 t+c_1{}^2+4 c_2}-t+c_1\right ) \\
\end{align*}