29.33.11 problem 973
Internal
problem
ID
[5550]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
33
Problem
number
:
973
Date
solved
:
Monday, January 27, 2025 at 11:44:57 AM
CAS
classification
:
[[_1st_order, _with_linear_symmetries]]
\begin{align*} y^{2} {y^{\prime }}^{2}-6 x^{3} y^{\prime }+4 x^{2} y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.171 (sec). Leaf size: 105
dsolve(y(x)^2*diff(y(x),x)^2-6*x^3*diff(y(x),x)+4*x^2*y(x) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {18^{{1}/{3}} x^{{4}/{3}}}{2} \\
y \left (x \right ) &= -\frac {18^{{1}/{3}} x^{{4}/{3}} \left (1+i \sqrt {3}\right )}{4} \\
y \left (x \right ) &= \frac {18^{{1}/{3}} x^{{4}/{3}} \left (i \sqrt {3}-1\right )}{4} \\
y \left (x \right ) &= 0 \\
y \left (x \right ) &= \operatorname {RootOf}\left (-4 \ln \left (x \right )-3 \left (\int _{}^{\textit {\_Z}}\frac {4 \textit {\_a}^{3}+3 \sqrt {-4 \textit {\_a}^{3}+9}-9}{\textit {\_a} \left (4 \textit {\_a}^{3}-9\right )}d \textit {\_a} \right )+4 c_{1} \right ) x^{{4}/{3}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 2.384 (sec). Leaf size: 235
DSolve[y[x]^2 (D[y[x],x])^2-6 x^3 D[y[x],x]+4 x^2 y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\frac {3}{4} \log (y(x))-\frac {\sqrt {9 x^6-4 x^2 y(x)^3} \text {arctanh}\left (\frac {3 x^2}{\sqrt {9 x^4-4 y(x)^3}+2 i y(x)^{3/2}}\right )}{x \sqrt {9 x^4-4 y(x)^3}}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {3}{4} \log (y(x))+\frac {\sqrt {9 x^6-4 x^2 y(x)^3} \text {arctanh}\left (\frac {3 x^2}{\sqrt {9 x^4-4 y(x)^3}+2 i y(x)^{3/2}}\right )}{x \sqrt {9 x^4-4 y(x)^3}}&=c_1,y(x)\right ] \\
y(x)\to \left (-\frac {3}{2}\right )^{2/3} x^{4/3} \\
y(x)\to \left (\frac {3}{2}\right )^{2/3} x^{4/3} \\
y(x)\to -\sqrt [3]{-1} \left (\frac {3}{2}\right )^{2/3} x^{4/3} \\
\end{align*}