23.1.652 problem 647
Internal
problem
ID
[5259]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
647
Date
solved
:
Tuesday, September 30, 2025 at 12:03:28 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational]
\begin{align*} \left (x \left (a -x^{2}-y^{2}\right )+y\right ) y^{\prime }+x -\left (a -x^{2}-y^{2}\right ) y&=0 \end{align*}
✓ Maple. Time used: 0.148 (sec). Leaf size: 37
ode:=(x*(a-x^2-y(x)^2)+y(x))*diff(y(x),x)+x-(a-x^2-y(x)^2)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = x \cot \left (\operatorname {RootOf}\left (2 c_1 a -2 \textit {\_Z} a +\ln \left (-\frac {x^{2}}{a \sin \left (\textit {\_Z} \right )^{2}-x^{2}}\right )\right )\right )
\]
✓ Mathematica. Time used: 0.221 (sec). Leaf size: 627
ode=(x*(a-x^2-y[x]^2)+y[x])*D[y[x],x]+x-(a-x^2-y[x]^2)*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (-\exp \left (\int _1^{x^2+K[3]^2}\left (\frac {1}{a-K[1]}-\frac {1}{K[1]}\right )dK[1]\right ) x^3+a \exp \left (\int _1^{x^2+K[3]^2}\left (\frac {1}{a-K[1]}-\frac {1}{K[1]}\right )dK[1]\right ) x-\exp \left (\int _1^{x^2+K[3]^2}\left (\frac {1}{a-K[1]}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^2 x+\exp \left (\int _1^{x^2+K[3]^2}\left (\frac {1}{a-K[1]}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]-\int _1^x\left (\exp \left (\int _1^{K[2]^2+K[3]^2}\left (\frac {1}{a-K[1]}-\frac {1}{K[1]}\right )dK[1]\right ) K[2]^2+2 \exp \left (\int _1^{K[2]^2+K[3]^2}\left (\frac {1}{a-K[1]}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^2 \left (\frac {1}{-K[2]^2-K[3]^2+a}-\frac {1}{K[2]^2+K[3]^2}\right ) K[2]^2+2 \exp \left (\int _1^{K[2]^2+K[3]^2}\left (\frac {1}{a-K[1]}-\frac {1}{K[1]}\right )dK[1]\right ) K[3] \left (\frac {1}{-K[2]^2-K[3]^2+a}-\frac {1}{K[2]^2+K[3]^2}\right ) K[2]+2 \exp \left (\int _1^{K[2]^2+K[3]^2}\left (\frac {1}{a-K[1]}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^2-\exp \left (\int _1^{K[2]^2+K[3]^2}\left (\frac {1}{a-K[1]}-\frac {1}{K[1]}\right )dK[1]\right ) \left (a-K[3]^2\right )-2 \exp \left (\int _1^{K[2]^2+K[3]^2}\left (\frac {1}{a-K[1]}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^2 \left (a-K[3]^2\right ) \left (\frac {1}{-K[2]^2-K[3]^2+a}-\frac {1}{K[2]^2+K[3]^2}\right )\right )dK[2]\right )dK[3]+\int _1^x\left (\exp \left (\int _1^{K[2]^2+y(x)^2}\left (\frac {1}{a-K[1]}-\frac {1}{K[1]}\right )dK[1]\right ) y(x) K[2]^2+\exp \left (\int _1^{K[2]^2+y(x)^2}\left (\frac {1}{a-K[1]}-\frac {1}{K[1]}\right )dK[1]\right ) K[2]-\exp \left (\int _1^{K[2]^2+y(x)^2}\left (\frac {1}{a-K[1]}-\frac {1}{K[1]}\right )dK[1]\right ) y(x) \left (a-y(x)^2\right )\right )dK[2]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(x + (x*(a - x**2 - y(x)**2) + y(x))*Derivative(y(x), x) - (a - x**2 - y(x)**2)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out