75.28.1 problem 787
Internal
problem
ID
[17177]
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
21.
Finding
integrable
combinations.
Exercises
page
219
Problem
number
:
787
Date
solved
:
Monday, March 31, 2025 at 03:43:56 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )^{2}+y \left (t \right )^{2}\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right ) y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.568 (sec). Leaf size: 65
ode:=[diff(x(t),t) = x(t)^2+y(t)^2, diff(y(t),t) = 2*x(t)*y(t)];
dsolve(ode);
\begin{align*}
\left [\{y \left (t \right ) = 0\}, \left \{x \left (t \right ) &= \frac {1}{-t +c_1}\right \}\right ] \\
\left [\left \{y \left (t \right ) &= \frac {4 c_1}{c_1^{2} c_2^{2}+2 c_1^{2} c_2 t +c_1^{2} t^{2}-16}\right \}, \left \{x \left (t \right ) = \frac {\frac {d}{d t}y \left (t \right )}{2 y \left (t \right )}\right \}\right ] \\
\end{align*}
✓ Mathematica. Time used: 0.481 (sec). Leaf size: 518
ode={D[x[t],t]==x[t]^2+y[t]^2,D[y[t],t]==-2*x[t]*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {\sqrt {3 c_1-\text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {\text {$\#$1}^3}{3 c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {\text {$\#$1}^3}{3 c_1}\right )}{\sqrt {-\text {$\#$1}^3+3 c_1}}\&\right ]\left [-\frac {2 t}{\sqrt {3}}+c_2\right ]{}^3}}{\sqrt {3} \sqrt {\text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {\text {$\#$1}^3}{3 c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {\text {$\#$1}^3}{3 c_1}\right )}{\sqrt {-\text {$\#$1}^3+3 c_1}}\&\right ]\left [-\frac {2 t}{\sqrt {3}}+c_2\right ]}} \\
y(t)\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {\text {$\#$1}^3}{3 c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {\text {$\#$1}^3}{3 c_1}\right )}{\sqrt {-\text {$\#$1}^3+3 c_1}}\&\right ]\left [-\frac {2 t}{\sqrt {3}}+c_2\right ] \\
x(t)\to -\frac {\sqrt {3 c_1-\text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {\text {$\#$1}^3}{3 c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {\text {$\#$1}^3}{3 c_1}\right )}{\sqrt {-\text {$\#$1}^3+3 c_1}}\&\right ]\left [\frac {2 t}{\sqrt {3}}+c_2\right ]{}^3}}{\sqrt {3} \sqrt {\text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {\text {$\#$1}^3}{3 c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {\text {$\#$1}^3}{3 c_1}\right )}{\sqrt {-\text {$\#$1}^3+3 c_1}}\&\right ]\left [\frac {2 t}{\sqrt {3}}+c_2\right ]}} \\
y(t)\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {\text {$\#$1}^3}{3 c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {\text {$\#$1}^3}{3 c_1}\right )}{\sqrt {-\text {$\#$1}^3+3 c_1}}\&\right ]\left [\frac {2 t}{\sqrt {3}}+c_2\right ] \\
\end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t)**2 - y(t)**2 + Derivative(x(t), t),0),Eq(-2*x(t)*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)