23.1.684 problem 679
Internal
problem
ID
[5291]
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
:
679
Date
solved
:
Tuesday, September 30, 2025 at 12:07:32 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (x^{3}-y^{3}\right ) y^{\prime }+x^{2} y&=0 \end{align*}
✓ Maple. Time used: 0.216 (sec). Leaf size: 199
ode:=(x^3-y(x)^3)*diff(y(x),x)+x^2*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (c_1 \,x^{3}-\sqrt {c_1^{2} x^{6}+1}\right )^{{1}/{3}}}{c_1^{{1}/{3}}} \\
y &= \frac {\left (c_1 \,x^{3}+\sqrt {c_1^{2} x^{6}+1}\right )^{{1}/{3}}}{c_1^{{1}/{3}}} \\
y &= -\frac {\left (c_1 \,x^{3}-\sqrt {c_1^{2} x^{6}+1}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 c_1^{{1}/{3}}} \\
y &= \frac {\left (c_1 \,x^{3}-\sqrt {c_1^{2} x^{6}+1}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 c_1^{{1}/{3}}} \\
y &= -\frac {\left (c_1 \,x^{3}+\sqrt {c_1^{2} x^{6}+1}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 c_1^{{1}/{3}}} \\
y &= \frac {\left (c_1 \,x^{3}+\sqrt {c_1^{2} x^{6}+1}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 c_1^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 5.113 (sec). Leaf size: 352
ode=(x^3-y[x]^3)D[y[x],x]+x^2 y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}}\\ y(x)&\to \sqrt [3]{x^3+\sqrt {x^6-e^{6 c_1}}}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{x^3+\sqrt {x^6-e^{6 c_1}}}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{x^3+\sqrt {x^6-e^{6 c_1}}}\\ y(x)&\to 0\\ y(x)&\to \sqrt [3]{x^3-\sqrt {x^6}}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{x^3-\sqrt {x^6}}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{x^3-\sqrt {x^6}}\\ y(x)&\to \sqrt [3]{\sqrt {x^6}+x^3}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{\sqrt {x^6}+x^3}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{\sqrt {x^6}+x^3} \end{align*}
✓ Sympy. Time used: 7.559 (sec). Leaf size: 146
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*y(x) + (x**3 - y(x)**3)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{x^{3} - \sqrt {C_{1} + x^{6}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{x^{3} + \sqrt {C_{1} + x^{6}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{x^{3} - \sqrt {C_{1} + x^{6}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{x^{3} + \sqrt {C_{1} + x^{6}}}}{2}, \ y{\left (x \right )} = \sqrt [3]{x^{3} - \sqrt {C_{1} + x^{6}}}, \ y{\left (x \right )} = \sqrt [3]{x^{3} + \sqrt {C_{1} + x^{6}}}\right ]
\]