23.2.253 problem 266
Internal
problem
ID
[5608]
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
:
266
Date
solved
:
Tuesday, September 30, 2025 at 01:11:43 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 9 x y^{4} {y^{\prime }}^{2}-3 y^{5} y^{\prime }-a&=0 \end{align*}
✓ Maple. Time used: 0.125 (sec). Leaf size: 279
ode:=9*x*y(x)^4*diff(y(x),x)^2-3*y(x)^5*diff(y(x),x)-a = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 2^{{1}/{3}} \left (-a x \right )^{{1}/{6}} \\
y &= -2^{{1}/{3}} \left (-a x \right )^{{1}/{6}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} \left (-a x \right )^{{1}/{6}}}{2} \\
y &= \frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (-a x \right )^{{1}/{6}}}{2} \\
y &= -\frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (-a x \right )^{{1}/{6}}}{2} \\
y &= \frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} \left (-a x \right )^{{1}/{6}}}{2} \\
y &= \frac {\left (a \left (c_1 -x \right )^{2} c_1^{5}\right )^{{1}/{6}}}{c_1} \\
y &= -\frac {\left (a \left (c_1 -x \right )^{2} c_1^{5}\right )^{{1}/{6}}}{c_1} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (a \left (c_1 -x \right )^{2} c_1^{5}\right )^{{1}/{6}}}{2 c_1} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (a \left (c_1 -x \right )^{2} c_1^{5}\right )^{{1}/{6}}}{2 c_1} \\
y &= -\frac {\left (i \sqrt {3}-1\right ) \left (a \left (c_1 -x \right )^{2} c_1^{5}\right )^{{1}/{6}}}{2 c_1} \\
y &= \frac {\left (1+i \sqrt {3}\right ) \left (a \left (c_1 -x \right )^{2} c_1^{5}\right )^{{1}/{6}}}{2 c_1} \\
\end{align*}
✓ Mathematica. Time used: 0.927 (sec). Leaf size: 358
ode=9 x y[x]^4 (D[y[x],x])^2 -3 y[x]^5 D[y[x],x]-a==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\sqrt [3]{-\frac {1}{2}} e^{-\frac {c_1}{6}} \sqrt [3]{-4 a x+e^{c_1}}\\ y(x)&\to \frac {e^{-\frac {c_1}{6}} \sqrt [3]{-4 a x+e^{c_1}}}{\sqrt [3]{2}}\\ y(x)&\to \frac {(-1)^{2/3} e^{-\frac {c_1}{6}} \sqrt [3]{-4 a x+e^{c_1}}}{\sqrt [3]{2}}\\ y(x)&\to -\sqrt [3]{-\frac {1}{2}} \sqrt [3]{-e^{-\frac {c_1}{2}} \left (-4 a x+e^{c_1}\right )}\\ y(x)&\to \frac {\sqrt [3]{e^{-\frac {c_1}{2}} \left (4 a x-e^{c_1}\right )}}{\sqrt [3]{2}}\\ y(x)&\to \frac {(-1)^{2/3} \sqrt [3]{-e^{-\frac {c_1}{2}} \left (-4 a x+e^{c_1}\right )}}{\sqrt [3]{2}}\\ y(x)&\to -i \sqrt [3]{2} \sqrt [6]{a} \sqrt [6]{x}\\ y(x)&\to i \sqrt [3]{2} \sqrt [6]{a} \sqrt [6]{x}\\ y(x)&\to -\sqrt [6]{-1} \sqrt [3]{2} \sqrt [6]{a} \sqrt [6]{x}\\ y(x)&\to \sqrt [6]{-1} \sqrt [3]{2} \sqrt [6]{a} \sqrt [6]{x}\\ y(x)&\to -(-1)^{5/6} \sqrt [3]{2} \sqrt [6]{a} \sqrt [6]{x}\\ y(x)&\to (-1)^{5/6} \sqrt [3]{2} \sqrt [6]{a} \sqrt [6]{x} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a + 9*x*y(x)**4*Derivative(y(x), x)**2 - 3*y(x)**5*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out