89.32.16 problem 19
Internal
problem
ID
[24976]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
16.
Equations
of
order
one
and
higher
degree.
Miscellaneous
Exercises
at
page
246
Problem
number
:
19
Date
solved
:
Thursday, October 02, 2025 at 11:45:15 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 9 x y^{4} {y^{\prime }}^{2}-3 y^{5} y^{\prime }-1&=0 \end{align*}
✓ Maple. Time used: 0.289 (sec). Leaf size: 267
ode:=9*x*y(x)^4*diff(y(x),x)^2-3*y(x)^5*diff(y(x),x)-1 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 2^{{1}/{3}} \left (-x \right )^{{1}/{6}} \\
y &= -2^{{1}/{3}} \left (-x \right )^{{1}/{6}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} \left (-x \right )^{{1}/{6}}}{2} \\
y &= \frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (-x \right )^{{1}/{6}}}{2} \\
y &= -\frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (-x \right )^{{1}/{6}}}{2} \\
y &= \frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} \left (-x \right )^{{1}/{6}}}{2} \\
y &= \frac {\left (\left (c_1 -x \right )^{2} c_1^{5}\right )^{{1}/{6}}}{c_1} \\
y &= -\frac {\left (\left (c_1 -x \right )^{2} c_1^{5}\right )^{{1}/{6}}}{c_1} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (\left (c_1 -x \right )^{2} c_1^{5}\right )^{{1}/{6}}}{2 c_1} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (\left (c_1 -x \right )^{2} c_1^{5}\right )^{{1}/{6}}}{2 c_1} \\
y &= -\frac {\left (i \sqrt {3}-1\right ) \left (\left (c_1 -x \right )^{2} c_1^{5}\right )^{{1}/{6}}}{2 c_1} \\
y &= \frac {\left (1+i \sqrt {3}\right ) \left (\left (c_1 -x \right )^{2} c_1^{5}\right )^{{1}/{6}}}{2 c_1} \\
\end{align*}
✓ Mathematica. Time used: 0.71 (sec). Leaf size: 322
ode=9*x*y[x]^4*D[y[x],x]^2-3*y[x]^5*D[y[x],x]-1==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 x+e^{c_1}}\\ y(x)&\to \frac {e^{-\frac {c_1}{6}} \sqrt [3]{-4 x+e^{c_1}}}{\sqrt [3]{2}}\\ y(x)&\to \frac {(-1)^{2/3} e^{-\frac {c_1}{6}} \sqrt [3]{-4 x+e^{c_1}}}{\sqrt [3]{2}}\\ y(x)&\to -\sqrt [3]{-\frac {1}{2}} \sqrt [3]{-e^{-\frac {c_1}{2}} \left (-4 x+e^{c_1}\right )}\\ y(x)&\to \frac {\sqrt [3]{e^{-\frac {c_1}{2}} \left (4 x-e^{c_1}\right )}}{\sqrt [3]{2}}\\ y(x)&\to \frac {(-1)^{2/3} \sqrt [3]{-e^{-\frac {c_1}{2}} \left (-4 x+e^{c_1}\right )}}{\sqrt [3]{2}}\\ y(x)&\to -i \sqrt [3]{2} \sqrt [6]{x}\\ y(x)&\to i \sqrt [3]{2} \sqrt [6]{x}\\ y(x)&\to -\sqrt [6]{-1} \sqrt [3]{2} \sqrt [6]{x}\\ y(x)&\to \sqrt [6]{-1} \sqrt [3]{2} \sqrt [6]{x}\\ y(x)&\to -(-1)^{5/6} \sqrt [3]{2} \sqrt [6]{x}\\ y(x)&\to (-1)^{5/6} \sqrt [3]{2} \sqrt [6]{x} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(9*x*y(x)**4*Derivative(y(x), x)**2 - 3*y(x)**5*Derivative(y(x), x) - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out