75.26.8 problem 775
Internal
problem
ID
[17165]
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
:
775
Date
solved
:
Monday, March 31, 2025 at 03:43:40 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 {t +x \left (t \right )}{y \left (t \right )+x \left (t \right )} \end{align*}
✓ Maple. Time used: 0.822 (sec). Leaf size: 1775
ode:=[diff(x(t),t) = (t+y(t))/(x(t)+y(t)), diff(y(t),t) = (t+x(t))/(x(t)+y(t))];
dsolve(ode);
\begin{align*}
\text {Expression too large to display} \\
\left [\left \{x \left (t \right ) &= \frac {t c_2 +\frac {\left (8 c_2^{3} t^{3}+4 \sqrt {4 c_2^{3} t^{3}+c_1^{2}}\, c_1 +4 c_1^{2}\right )^{{1}/{3}}}{2 c_1}+\frac {2 t^{2} c_2^{2}}{c_1 \left (8 c_2^{3} t^{3}+4 \sqrt {4 c_2^{3} t^{3}+c_1^{2}}\, c_1 +4 c_1^{2}\right )^{{1}/{3}}}+\frac {t c_2}{c_1}}{c_2}, x \left (t \right ) = \frac {t c_2 -\frac {\left (8 c_2^{3} t^{3}+4 \sqrt {4 c_2^{3} t^{3}+c_1^{2}}\, c_1 +4 c_1^{2}\right )^{{1}/{3}}}{4 c_1}-\frac {t^{2} c_2^{2}}{c_1 \left (8 c_2^{3} t^{3}+4 \sqrt {4 c_2^{3} t^{3}+c_1^{2}}\, c_1 +4 c_1^{2}\right )^{{1}/{3}}}+\frac {t c_2}{c_1}-\frac {i \sqrt {3}\, \left (\frac {\left (8 c_2^{3} t^{3}+4 \sqrt {4 c_2^{3} t^{3}+c_1^{2}}\, c_1 +4 c_1^{2}\right )^{{1}/{3}}}{2 c_1}-\frac {2 t^{2} c_2^{2}}{c_1 \left (8 c_2^{3} t^{3}+4 \sqrt {4 c_2^{3} t^{3}+c_1^{2}}\, c_1 +4 c_1^{2}\right )^{{1}/{3}}}\right )}{2}}{c_2}, x \left (t \right ) = \frac {t c_2 -\frac {\left (8 c_2^{3} t^{3}+4 \sqrt {4 c_2^{3} t^{3}+c_1^{2}}\, c_1 +4 c_1^{2}\right )^{{1}/{3}}}{4 c_1}-\frac {t^{2} c_2^{2}}{c_1 \left (8 c_2^{3} t^{3}+4 \sqrt {4 c_2^{3} t^{3}+c_1^{2}}\, c_1 +4 c_1^{2}\right )^{{1}/{3}}}+\frac {t c_2}{c_1}+\frac {i \sqrt {3}\, \left (\frac {\left (8 c_2^{3} t^{3}+4 \sqrt {4 c_2^{3} t^{3}+c_1^{2}}\, c_1 +4 c_1^{2}\right )^{{1}/{3}}}{2 c_1}-\frac {2 t^{2} c_2^{2}}{c_1 \left (8 c_2^{3} t^{3}+4 \sqrt {4 c_2^{3} t^{3}+c_1^{2}}\, c_1 +4 c_1^{2}\right )^{{1}/{3}}}\right )}{2}}{c_2}\right \}, \left \{y \left (t \right ) = \frac {-\left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )+t}{\frac {d}{d t}x \left (t \right )-1}\right \}\right ] \\
\end{align*}
✗ Mathematica
ode={D[x[t],t]==(t+y[t])/(y[t]+x[t]),D[y[t],t]==(x[t]+t)/(y[t]+x[t])};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
Not solved
✗ Sympy
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq((-t - y(t))/(x(t) + y(t)) + Derivative(x(t), t),0),Eq((-t - x(t))/(x(t) + y(t)) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
NotImplementedError :