29.24.14 problem 676
Internal
problem
ID
[5267]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
24
Problem
number
:
676
Date
solved
:
Tuesday, March 04, 2025 at 08:58:46 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: 1.217 (sec). Leaf size: 381
ode:=(x^3-y(x)^3)*diff(y(x),x)+x^2*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \frac {x}{{\left (\left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\
y \left (x \right ) &= \frac {x}{{\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) x^{3} c_{1} \right )}^{{1}/{3}}} \\
y \left (x \right ) &= \frac {4 x}{\left (1+i \sqrt {3}\right )^{2} {\left (\left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\
y \left (x \right ) &= \frac {4 x}{\left (i \sqrt {3}-1\right )^{2} {\left (\left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\
y \left (x \right ) &= \frac {4 x}{\left (i \sqrt {3}-1\right )^{2} {\left (\left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\
y \left (x \right ) &= \frac {4 x}{\left (1+i \sqrt {3}\right )^{2} {\left (\left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\
y \left (x \right ) &= \frac {4 x}{\left (1+i \sqrt {3}\right )^{2} {\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) x^{3} c_{1} \right )}^{{1}/{3}}} \\
y \left (x \right ) &= \frac {4 x}{\left (i \sqrt {3}-1\right )^{2} {\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) x^{3} c_{1} \right )}^{{1}/{3}}} \\
y \left (x \right ) &= \frac {4 x}{\left (i \sqrt {3}-1\right )^{2} {\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) x^{3} c_{1} \right )}^{{1}/{3}}} \\
y \left (x \right ) &= \frac {4 x}{\left (1+i \sqrt {3}\right )^{2} {\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) x^{3} c_{1} \right )}^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 6.764 (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: 12.133 (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)
\[
\left [ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{x^{3} - \sqrt {C_{1} + x^{6}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{x^{3} + \sqrt {C_{1} + x^{6}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{x^{3} - \sqrt {C_{1} + x^{6}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{x^{3} + \sqrt {C_{1} + x^{6}}}}{2}, \ y{\left (x \right )} = \sqrt [3]{x^{3} - \sqrt {C_{1} + x^{6}}}, \ y{\left (x \right )} = \sqrt [3]{x^{3} + \sqrt {C_{1} + x^{6}}}\right ]
\]