89.31.2 problem 2
Internal
problem
ID
[24947]
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.
Exercises
at
page
246
Problem
number
:
2
Date
solved
:
Thursday, October 02, 2025 at 11:36:27 PM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} 9 {y^{\prime }}^{2} x +3 y y^{\prime }+y^{8}&=0 \end{align*}
✓ Maple. Time used: 0.024 (sec). Leaf size: 169
ode:=9*x*diff(y(x),x)^2+3*y(x)*diff(y(x),x)+y(x)^8 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {2^{{2}/{3}}}{2 x^{{1}/{6}}} \\
y &= \frac {2^{{2}/{3}}}{2 x^{{1}/{6}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) 2^{{2}/{3}}}{4 x^{{1}/{6}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) 2^{{2}/{3}}}{4 x^{{1}/{6}}} \\
y &= -\frac {\left (i \sqrt {3}-1\right ) 2^{{2}/{3}}}{4 x^{{1}/{6}}} \\
y &= \frac {\left (1+i \sqrt {3}\right ) 2^{{2}/{3}}}{4 x^{{1}/{6}}} \\
y &= 0 \\
y &= \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_1 +6 \int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} \sqrt {-4 \textit {\_a}^{6}+1}}d \textit {\_a} \right )}{x^{{1}/{6}}} \\
y &= \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_1 -6 \int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} \sqrt {-4 \textit {\_a}^{6}+1}}d \textit {\_a} \right )}{x^{{1}/{6}}} \\
\end{align*}
✓ Mathematica. Time used: 0.398 (sec). Leaf size: 306
ode=9*x*D[y[x],x]^2+3*y[x]*D[y[x],x]+y[x]^8==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {\sqrt [3]{-2} e^{\frac {c_1}{6}}}{\sqrt [3]{4 x+e^{c_1}}}\\ y(x)&\to \frac {\sqrt [3]{-2} e^{\frac {c_1}{6}}}{\sqrt [3]{4 x+e^{c_1}}}\\ y(x)&\to -\frac {\sqrt [3]{2} e^{\frac {c_1}{6}}}{\sqrt [3]{4 x+e^{c_1}}}\\ y(x)&\to \frac {\sqrt [3]{2} e^{\frac {c_1}{6}}}{\sqrt [3]{4 x+e^{c_1}}}\\ y(x)&\to -\frac {(-1)^{2/3} \sqrt [3]{2} e^{\frac {c_1}{6}}}{\sqrt [3]{4 x+e^{c_1}}}\\ y(x)&\to \frac {(-1)^{2/3} \sqrt [3]{2} e^{\frac {c_1}{6}}}{\sqrt [3]{4 x+e^{c_1}}}\\ y(x)&\to 0\\ y(x)&\to -\frac {\sqrt [3]{-\frac {1}{2}}}{\sqrt [6]{x}}\\ y(x)&\to \frac {\sqrt [3]{-\frac {1}{2}}}{\sqrt [6]{x}}\\ y(x)&\to -\frac {1}{\sqrt [3]{2} \sqrt [6]{x}}\\ y(x)&\to \frac {1}{\sqrt [3]{2} \sqrt [6]{x}}\\ y(x)&\to -\frac {(-1)^{2/3}}{\sqrt [3]{2} \sqrt [6]{x}}\\ y(x)&\to \frac {(-1)^{2/3}}{\sqrt [3]{2} \sqrt [6]{x}} \end{align*}
✓ Sympy. Time used: 0.818 (sec). Leaf size: 99
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(9*x*Derivative(y(x), x)**2 + y(x)**8 + 3*y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} 2 \operatorname {acosh}{\left (\frac {1}{2 \sqrt {x} y^{3}{\left (x \right )}} \right )} & \text {for}\: \frac {1}{\left |{x y^{6}{\left (x \right )}}\right |} > 4 \\- 2 i \operatorname {asin}{\left (\frac {1}{2 \sqrt {x} y^{3}{\left (x \right )}} \right )} & \text {otherwise} \end {cases} = C_{1} + \log {\left (x \right )}, \ \begin {cases} - 2 \operatorname {acosh}{\left (\frac {1}{2 \sqrt {x} y^{3}{\left (x \right )}} \right )} & \text {for}\: \frac {1}{\left |{x y^{6}{\left (x \right )}}\right |} > 4 \\2 i \operatorname {asin}{\left (\frac {1}{2 \sqrt {x} y^{3}{\left (x \right )}} \right )} & \text {otherwise} \end {cases} = C_{1} + \log {\left (x \right )}\right ]
\]