23.2.242 problem 248
Internal
problem
ID
[5597]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
2.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
SECOND
OR
HIGHER
DEGREE,
page
278
Problem
number
:
248
Date
solved
:
Tuesday, September 30, 2025 at 01:10:50 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational]
\begin{align*} 9 y^{2} {y^{\prime }}^{2}-3 x y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.105 (sec). Leaf size: 112
ode:=9*y(x)^2*diff(y(x),x)^2-3*x*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}}}{2} \\
y &= -\frac {2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4} \\
y &= \frac {2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}} \left (-1+i \sqrt {3}\right )}{4} \\
y &= 0 \\
y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+3 \int _{}^{\textit {\_Z}}-\frac {4 \textit {\_a}^{3}-\sqrt {-4 \textit {\_a}^{3}+1}-1}{\textit {\_a} \left (4 \textit {\_a}^{3}-1\right )}d \textit {\_a} +2 c_1 \right ) x^{{2}/{3}} \\
\end{align*}
✓ Mathematica. Time used: 0.845 (sec). Leaf size: 418
ode=9 y[x]^2 (D[y[x],x])^2 -3 x D[y[x],x]+y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\sqrt [3]{-2} \sqrt [3]{e^{c_1} \left (x-2 e^{c_1}\right )}\\ y(x)&\to \sqrt [3]{2} \sqrt [3]{e^{c_1} \left (x-2 e^{c_1}\right )}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{2} \sqrt [3]{e^{c_1} \left (x-2 e^{c_1}\right )}\\ y(x)&\to \left (\cosh \left (\frac {c_1}{3}\right )+\sinh \left (\frac {c_1}{3}\right )\right ) \sqrt [3]{-i x+\cosh (c_1)+\sinh (c_1)}\\ y(x)&\to \frac {1}{2} i \left (\sqrt {3}+i\right ) \left (\cosh \left (\frac {c_1}{3}\right )+\sinh \left (\frac {c_1}{3}\right )\right ) \sqrt [3]{-i x+\cosh (c_1)+\sinh (c_1)}\\ y(x)&\to -\frac {1}{2} i \left (\sqrt {3}-i\right ) \left (\cosh \left (\frac {c_1}{3}\right )+\sinh \left (\frac {c_1}{3}\right )\right ) \sqrt [3]{-i x+\cosh (c_1)+\sinh (c_1)}\\ y(x)&\to \left (\cosh \left (\frac {c_1}{3}\right )+\sinh \left (\frac {c_1}{3}\right )\right ) \sqrt [3]{i x+\cosh (c_1)+\sinh (c_1)}\\ y(x)&\to \frac {1}{2} i \left (\sqrt {3}+i\right ) \left (\cosh \left (\frac {c_1}{3}\right )+\sinh \left (\frac {c_1}{3}\right )\right ) \sqrt [3]{i x+\cosh (c_1)+\sinh (c_1)}\\ y(x)&\to -\frac {1}{2} i \left (\sqrt {3}-i\right ) \left (\cosh \left (\frac {c_1}{3}\right )+\sinh \left (\frac {c_1}{3}\right )\right ) \sqrt [3]{i x+\cosh (c_1)+\sinh (c_1)}\\ y(x)&\to 0\\ y(x)&\to \left (-\frac {1}{2}\right )^{2/3} x^{2/3}\\ y(x)&\to \frac {x^{2/3}}{2^{2/3}}\\ y(x)&\to -\frac {\sqrt [3]{-1} x^{2/3}}{2^{2/3}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-3*x*Derivative(y(x), x) + 9*y(x)**2*Derivative(y(x), x)**2 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -(x + sqrt(x**2 - 4*y(x)**3))/(6*y(x)**2) + Derivative(y(x), x) cannot be solved by the factorable group method