29.23.25 problem 656
Internal
problem
ID
[5247]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
23
Problem
number
:
656
Date
solved
:
Monday, January 27, 2025 at 10:49:12 AM
CAS
classification
:
[_rational]
\begin{align*} 3 x \left (x +y^{2}\right ) y^{\prime }+x^{3}-3 y x -2 y^{3}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 356
dsolve(3*x*(x+y(x)^2)*diff(y(x),x)+x^3-3*x*y(x)-2*y(x)^3 = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {x^{3} \left (c_{1}^{2} x +2 c_{1} x^{2}+x^{3}+4\right )}\right )^{{2}/{3}}-4 x}{2 \left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {x^{3} \left (c_{1}^{2} x +2 c_{1} x^{2}+x^{3}+4\right )}\right )^{{1}/{3}}} \\
y \left (x \right ) &= -\frac {i \sqrt {3}\, \left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {x^{3} \left (c_{1}^{2} x +2 c_{1} x^{2}+x^{3}+4\right )}\right )^{{2}/{3}}+4 i \sqrt {3}\, x +\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {x^{3} \left (c_{1}^{2} x +2 c_{1} x^{2}+x^{3}+4\right )}\right )^{{2}/{3}}-4 x}{4 \left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {x^{3} \left (c_{1}^{2} x +2 c_{1} x^{2}+x^{3}+4\right )}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {i \sqrt {3}\, \left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {x^{3} \left (c_{1}^{2} x +2 c_{1} x^{2}+x^{3}+4\right )}\right )^{{2}/{3}}+4 i \sqrt {3}\, x -\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {x^{3} \left (c_{1}^{2} x +2 c_{1} x^{2}+x^{3}+4\right )}\right )^{{2}/{3}}+4 x}{4 \left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {x^{3} \left (c_{1}^{2} x +2 c_{1} x^{2}+x^{3}+4\right )}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 49.846 (sec). Leaf size: 362
DSolve[3 x(x+y[x]^2)D[y[x],x]+x^3-3 x y[x]-2 y[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{-x^3+c_1 x^2+\sqrt {x^3 \left (x^3-2 c_1 x^2+c_1{}^2 x+4\right )}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{-x^3+c_1 x^2+\sqrt {x^3 \left (x^3-2 c_1 x^2+c_1{}^2 x+4\right )}}} \\
y(x)\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \left (-x^3+c_1 x^2+\sqrt {x^3 \left (x^3-2 c_1 x^2+c_1{}^2 x+4\right )}\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x}{4 \sqrt [3]{-x^3+c_1 x^2+\sqrt {x^3 \left (x^3-2 c_1 x^2+c_1{}^2 x+4\right )}}} \\
y(x)\to \frac {\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) x-i 2^{2/3} \left (\sqrt {3}-i\right ) \left (-x^3+c_1 x^2+\sqrt {x^3 \left (x^3-2 c_1 x^2+c_1{}^2 x+4\right )}\right ){}^{2/3}}{4 \sqrt [3]{-x^3+c_1 x^2+\sqrt {x^3 \left (x^3-2 c_1 x^2+c_1{}^2 x+4\right )}}} \\
\end{align*}