29.29.20 problem 842
Internal
problem
ID
[5425]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
29
Problem
number
:
842
Date
solved
:
Monday, January 27, 2025 at 11:21:57 AM
CAS
classification
:
[[_1st_order, _with_linear_symmetries]]
\begin{align*} 9 {y^{\prime }}^{2}+3 x y^{4} y^{\prime }+y^{5}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.263 (sec). Leaf size: 100
dsolve(9*diff(y(x),x)^2+3*x*y(x)^4*diff(y(x),x)+y(x)^5 = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {2^{{2}/{3}}}{x^{{2}/{3}}} \\
y \left (x \right ) &= -\frac {2^{{2}/{3}} \left (1+i \sqrt {3}\right )}{2 x^{{2}/{3}}} \\
y \left (x \right ) &= \frac {2^{{2}/{3}} \left (i \sqrt {3}-1\right )}{2 x^{{2}/{3}}} \\
y \left (x \right ) &= 0 \\
y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-2 \ln \left (x \right )+3 \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}+\sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4\right )}-4}{\textit {\_a} \left (\textit {\_a}^{3}-4\right )}d \textit {\_a} \right )+2 c_{1} \right )}{x^{{2}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.769 (sec). Leaf size: 216
DSolve[9 (D[y[x],x])^2+3 x y[x]^4 D[y[x],x]+y[x]^5==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [-\frac {\sqrt {4-x^2 y(x)^3} y(x)^4 \text {arcsinh}\left (\frac {1}{2} x \sqrt {-y(x)^3}\right )}{\sqrt {-y(x)^3} \sqrt {y(x)^5 \left (x^2 y(x)^3-4\right )}}-\frac {3}{2} \log (y(x))&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {y(x)^4 \sqrt {4-x^2 y(x)^3} \text {arcsinh}\left (\frac {1}{2} x \sqrt {-y(x)^3}\right )}{\sqrt {-y(x)^3} \sqrt {y(x)^5 \left (x^2 y(x)^3-4\right )}}-\frac {3}{2} \log (y(x))&=c_1,y(x)\right ] \\
y(x)\to 0 \\
y(x)\to \frac {(-2)^{2/3}}{x^{2/3}} \\
y(x)\to \frac {2^{2/3}}{x^{2/3}} \\
y(x)\to -\frac {\sqrt [3]{-1} 2^{2/3}}{x^{2/3}} \\
\end{align*}