54.2.15 problem 18
Internal
problem
ID
[8547]
Book
:
Elementary
differential
equations.
Rainville,
Bedient,
Bedient.
Prentice
Hall.
NJ.
8th
edition.
1997.
Section
:
CHAPTER
16.
Nonlinear
equations.
Miscellaneous
Exercises.
Page
340
Problem
number
:
18
Date
solved
:
Monday, January 27, 2025 at 04:13:34 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*}
✓ Solution by Maple
Time used: 0.054 (sec). Leaf size: 259
dsolve(9*x*y(x)^4*diff(y(x),x)^2-3*y(x)^5*diff(y(x),x)-1=0,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 (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{c_{1}} \\
y &= -\frac {\left (\left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{c_{1}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (\left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{2 c_{1}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (\left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{2 c_{1}} \\
y &= -\frac {\left (i \sqrt {3}-1\right ) \left (\left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{2 c_{1}} \\
y &= \frac {\left (1+i \sqrt {3}\right ) \left (\left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{2 c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 3.107 (sec). Leaf size: 322
DSolve[9*x*y[x]^4*(D[y[x],x])^2-3*y[x]^5*D[y[x],x]-1==0,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*}