29.24.29 problem 692
Internal
problem
ID
[5282]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
24
Problem
number
:
692
Date
solved
:
Monday, January 27, 2025 at 11:03:47 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} x \left (x -y^{3}\right ) y^{\prime }&=\left (3 x +y^{3}\right ) y \end{align*}
✓ Solution by Maple
Time used: 0.120 (sec). Leaf size: 268
dsolve(x*(x-y(x)^3)*diff(y(x),x) = (3*x+y(x)^3)*y(x),y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{{2}/{3}}+3 \,{\mathrm e}^{\frac {8 c_{1}}{3}}}{3 x \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {-i \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{{2}/{3}} \sqrt {3}+3 i {\mathrm e}^{\frac {8 c_{1}}{3}} \sqrt {3}-\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{{2}/{3}}-3 \,{\mathrm e}^{\frac {8 c_{1}}{3}}}{6 x \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{{1}/{3}}} \\
y \left (x \right ) &= -\frac {-i \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{{2}/{3}} \sqrt {3}+3 i {\mathrm e}^{\frac {8 c_{1}}{3}} \sqrt {3}+\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{{2}/{3}}+3 \,{\mathrm e}^{\frac {8 c_{1}}{3}}}{6 x \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.202 (sec). Leaf size: 356
DSolve[x(x-y[x]^3)D[y[x],x]==(3 x+y[x]^3)y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {e^{\frac {8 c_1}{3}}}{\sqrt [3]{-27 x^7+3 \sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}}+\frac {\sqrt [3]{-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}}{3^{2/3} x^2} \\
y(x)\to \frac {\frac {i \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}\right ){}^{2/3}}{x^2}-\left (\sqrt {3}+3 i\right ) e^{\frac {8 c_1}{3}}}{2\ 3^{5/6} \sqrt [3]{-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}} \\
y(x)\to \frac {\frac {\left (-1-i \sqrt {3}\right ) \left (-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}\right ){}^{2/3}}{x^2}+i \sqrt [3]{3} \left (\sqrt {3}+i\right ) e^{\frac {8 c_1}{3}}}{2\ 3^{2/3} \sqrt [3]{-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}} \\
\end{align*}