54.2.3 problem 3

Internal problem ID [8535]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number : 3
Date solved : Monday, January 27, 2025 at 04:09:55 PM
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: 0.186 (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 &= \frac {2^{{2}/{3}}}{x^{{2}/{3}}} \\ y &= -\frac {2^{{2}/{3}} \left (1+i \sqrt {3}\right )}{2 x^{{2}/{3}}} \\ y &= \frac {2^{{2}/{3}} \left (i \sqrt {3}-1\right )}{2 x^{{2}/{3}}} \\ y &= 0 \\ y &= \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: 0.750 (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*}