29.33.32 problem 995
Internal
problem
ID
[5571]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
33
Problem
number
:
995
Date
solved
:
Monday, January 27, 2025 at 12:07:12 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*}
✓ Solution by Maple
Time used: 0.136 (sec). Leaf size: 112
dsolve(9*y(x)^2*diff(y(x),x)^2-3*x*diff(y(x),x)+y(x) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}}}{2} \\
y \left (x \right ) &= -\frac {2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4} \\
y \left (x \right ) &= \frac {2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4} \\
y \left (x \right ) &= 0 \\
y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+3 \left (\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} \right )+2 c_{1} \right ) x^{{2}/{3}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.498 (sec). Leaf size: 238
DSolve[9 y[x]^2 (D[y[x],x])^2 -3 x D[y[x],x]+y[x]==0,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 \frac {\sqrt [3]{-e^{c_1} \left (-2 x+e^{c_1}\right )}}{2^{2/3}} \\
y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{-e^{c_1} \left (-2 x+e^{c_1}\right )}}{2^{2/3}} \\
y(x)\to \left (-\frac {1}{2}\right )^{2/3} \sqrt [3]{-e^{c_1} \left (-2 x+e^{c_1}\right )} \\
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*}