23.1.663 problem 658
Internal
problem
ID
[5270]
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
:
658
Date
solved
:
Tuesday, September 30, 2025 at 12:06:16 PM
CAS
classification
:
[[_homogeneous, `class G`], _exact, _rational]
\begin{align*} x \left (x -3 y^{2}\right ) y^{\prime }+\left (2 x -y^{2}\right ) y&=0 \end{align*}
✓ Maple. Time used: 0.097 (sec). Leaf size: 242
ode:=x*(x-3*y(x)^2)*diff(y(x),x)+(2*x-y(x)^2)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {12^{{1}/{3}} \left (x^{3} 12^{{1}/{3}}+{\left (\left (\sqrt {-12 x^{5}+81 c_1^{2}}+9 c_1 \right ) x^{2}\right )}^{{2}/{3}}\right )}{6 x {\left (\left (\sqrt {-12 x^{5}+81 c_1^{2}}+9 c_1 \right ) x^{2}\right )}^{{1}/{3}}} \\
y &= \frac {\left (\left (-i \sqrt {3}-1\right ) {\left (\left (\sqrt {-12 x^{5}+81 c_1^{2}}+9 c_1 \right ) x^{2}\right )}^{{2}/{3}}+x^{3} 2^{{2}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )\right ) 3^{{1}/{3}} 2^{{2}/{3}}}{12 {\left (\left (\sqrt {-12 x^{5}+81 c_1^{2}}+9 c_1 \right ) x^{2}\right )}^{{1}/{3}} x} \\
y &= -\frac {3^{{1}/{3}} \left (\left (1-i \sqrt {3}\right ) {\left (\left (\sqrt {-12 x^{5}+81 c_1^{2}}+9 c_1 \right ) x^{2}\right )}^{{2}/{3}}+\left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) x^{3} 2^{{2}/{3}}\right ) 2^{{2}/{3}}}{12 {\left (\left (\sqrt {-12 x^{5}+81 c_1^{2}}+9 c_1 \right ) x^{2}\right )}^{{1}/{3}} x} \\
\end{align*}
✓ Mathematica. Time used: 40.843 (sec). Leaf size: 328
ode=x(x-3 y[x]^2)D[y[x],x]+(2 x-y[x]^2)y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {2 \sqrt [3]{3} x^3+\sqrt [3]{2} \left (9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{6^{2/3} x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}}\\ y(x)&\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^3+\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}}\\ y(x)&\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^3+\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x - 3*y(x)**2)*Derivative(y(x), x) + (2*x - y(x)**2)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out