23.1.701 problem 697
Internal
problem
ID
[5308]
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
:
697
Date
solved
:
Tuesday, September 30, 2025 at 12:28:59 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x \left (2 x^{3}-y^{3}\right ) y^{\prime }&=\left (x^{3}-2 y^{3}\right ) y \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 327
ode:=x*(2*x^3-y(x)^3)*diff(y(x),x) = (x^3-2*y(x)^3)*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (\frac {\left (-108+8 c_1^{3} x^{3}+12 \sqrt {-12 c_1^{3} x^{3}+81}\right )^{{1}/{3}}}{2}+\frac {2 c_1^{2} x^{2}}{\left (-108+8 c_1^{3} x^{3}+12 \sqrt {-12 c_1^{3} x^{3}+81}\right )^{{1}/{3}}}+c_1 x \right ) x}{3} \\
y &= -\frac {\left (-4 i \sqrt {3}\, c_1^{2} x^{2}+i \sqrt {3}\, \left (-108+8 c_1^{3} x^{3}+12 \sqrt {-12 c_1^{3} x^{3}+81}\right )^{{2}/{3}}+4 c_1^{2} x^{2}-4 c_1 x \left (-108+8 c_1^{3} x^{3}+12 \sqrt {-12 c_1^{3} x^{3}+81}\right )^{{1}/{3}}+\left (-108+8 c_1^{3} x^{3}+12 \sqrt {-12 c_1^{3} x^{3}+81}\right )^{{2}/{3}}\right ) x}{12 \left (-108+8 c_1^{3} x^{3}+12 \sqrt {-12 c_1^{3} x^{3}+81}\right )^{{1}/{3}}} \\
y &= \frac {\left (-108+8 c_1^{3} x^{3}+12 \sqrt {-12 c_1^{3} x^{3}+81}\right )^{{1}/{3}} x \left (i \sqrt {3}-1\right )}{12}-\frac {c_1 \left (i x c_1 \sqrt {3}+c_1 x -\left (-108+8 c_1^{3} x^{3}+12 \sqrt {-12 c_1^{3} x^{3}+81}\right )^{{1}/{3}}\right ) x^{2}}{3 \left (-108+8 c_1^{3} x^{3}+12 \sqrt {-12 c_1^{3} x^{3}+81}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 37.616 (sec). Leaf size: 542
ode=x(2 x^3-y[x]^3)D[y[x],x]==(x^3-2 y[x]^3)y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{3} \left (e^{c_1} x^2+\frac {\sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} e^{2 c_1} x^4}{\sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}\right )\\ y(x)&\to \frac {e^{c_1} x^2}{3}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}{6 \sqrt [3]{2}}-\frac {i \left (\sqrt {3}-i\right ) e^{2 c_1} x^4}{3\ 2^{2/3} \sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}\\ y(x)&\to \frac {e^{c_1} x^2}{3}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}{6 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) e^{2 c_1} x^4}{3\ 2^{2/3} \sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}\\ y(x)&\to \frac {\sqrt [3]{\sqrt {x^6}-x^3}}{\sqrt [3]{2}}\\ y(x)&\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {x^6}-x^3}}{2 \sqrt [3]{2}}\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {x^6}-x^3}}{2 \sqrt [3]{2}} \end{align*}
✓ Sympy. Time used: 16.598 (sec). Leaf size: 15
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(2*x**3 - y(x)**3)*Derivative(y(x), x) - (x**3 - 2*y(x)**3)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {C_{1} x}{6 \left (1 - \sqrt {3} i\right )}
\]