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
:
Sunday, March 30, 2025 at 07:50:40 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*}
✓ Maple. Time used: 0.040 (sec). Leaf size: 268
ode:=x*(x-y(x)^3)*diff(y(x),x) = (3*x+y(x)^3)*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \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 &= \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 &= -\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*}
✓ Mathematica. Time used: 60.202 (sec). Leaf size: 356
ode=x(x-y[x]^3)D[y[x],x]==(3 x+y[x]^3)y[x];
ic={};
DSolve[{ode,ic},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*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x - y(x)**3)*Derivative(y(x), x) - (3*x + y(x)**3)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out