29.24.33 problem 696
Internal
problem
ID
[5286]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
24
Problem
number
:
696
Date
solved
:
Tuesday, March 04, 2025 at 09:14:27 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x \left (x^{3}-2 y^{3}\right ) y^{\prime }&=\left (2 x^{3}-y^{3}\right ) y \end{align*}
✓ Maple. Time used: 0.035 (sec). Leaf size: 313
ode:=x*(x^3-2*y(x)^3)*diff(y(x),x) = (2*x^3-y(x)^3)*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \frac {12^{{1}/{3}} \left (x 12^{{1}/{3}} c_{1} +{\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{2}/{3}}\right )}{6 c_{1} {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{1}/{3}}} \\
y \left (x \right ) &= \frac {2^{{2}/{3}} \left (\left (-1-i \sqrt {3}\right ) {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} x c_{1} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )\right ) 3^{{1}/{3}}}{12 {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\
y \left (x \right ) &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (1-i \sqrt {3}\right ) {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) x c_{1} \right )}{12 {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\
\end{align*}
✓ Mathematica. Time used: 60.258 (sec). Leaf size: 331
ode=x(x^3-2 y[x]^3)D[y[x],x]==(2 x^3-y[x]^3)y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{2} \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}+2 \sqrt [3]{3} e^{c_1} x}{6^{2/3} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \\
y(x)\to \frac {i \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}+3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \\
y(x)\to \frac {\sqrt [3]{2} \sqrt [6]{3} \left (-1-i \sqrt {3}\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}-3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \\
\end{align*}
✓ Sympy. Time used: 6.979 (sec). Leaf size: 5
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x**3 - 2*y(x)**3)*Derivative(y(x), x) - (2*x**3 - y(x)**3)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} x
\]