29.24.31 problem 694
Internal
problem
ID
[5284]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
24
Problem
number
:
694
Date
solved
:
Monday, January 27, 2025 at 11:03:57 AM
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*}
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 323
dsolve(x*(2*x^3-y(x)^3)*diff(y(x),x) = (x^3-2*y(x)^3)*y(x),y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \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 x^{2} c_{1}^{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 \left (x \right ) &= -\frac {\left (-4 i \sqrt {3}\, c_{1}^{2} x^{2}+i \left (-108+8 c_{1}^{3} x^{3}+12 \sqrt {-12 c_{1}^{3} x^{3}+81}\right )^{{2}/{3}} \sqrt {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 \left (x \right ) &= \frac {x \left (i \sqrt {3}-1\right ) \left (-108+8 c_{1}^{3} x^{3}+12 \sqrt {-12 c_{1}^{3} x^{3}+81}\right )^{{1}/{3}}}{12}-\frac {x^{2} \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 ) c_{1}}{3 \left (-108+8 c_{1}^{3} x^{3}+12 \sqrt {-12 c_{1}^{3} x^{3}+81}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 45.532 (sec). Leaf size: 542
DSolve[x(2 x^3-y[x]^3)D[y[x],x]==(x^3-2 y[x]^3)y[x],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*}