Internal problem ID [6867]
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.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
\[ \boxed {9 {y^{\prime }}^{2}+3 y^{4} y^{\prime } x +y^{5}=0} \]
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 109
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 {4^{\frac {1}{3}}}{x^{\frac {2}{3}}} y \left (x \right ) = -\frac {4^{\frac {1}{3}}}{2 x^{\frac {2}{3}}}-\frac {i \sqrt {3}\, 4^{\frac {1}{3}}}{2 x^{\frac {2}{3}}} y \left (x \right ) = -\frac {4^{\frac {1}{3}}}{2 x^{\frac {2}{3}}}+\frac {i \sqrt {3}\, 4^{\frac {1}{3}}}{2 x^{\frac {2}{3}}} y \left (x \right ) = 0 y \left (x \right ) = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\frac {3 \textit {\_a}^{3}}{2}+\frac {3 \sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4\right )}}{2}-6}{\textit {\_a} \left (\textit {\_a}^{3}-4\right )}d \textit {\_a} +c_{1} \right )}{x^{\frac {2}{3}}} \end{align*}
✓ Solution by Mathematica
Time used: 1.025 (sec). Leaf size: 212
DSolve[9*(y'[x])^2+3*x*y[x]^4*y'[x]+y[x]^5==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {\sqrt {x^2 y(x)^3-4} y(x)^{5/2} \text {arctanh}\left (\frac {x y(x)^{3/2}}{\sqrt {x^2 y(x)^3-4}}\right )}{\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)^{5/2} \sqrt {x^2 y(x)^3-4} \text {arctanh}\left (\frac {x y(x)^{3/2}}{\sqrt {x^2 y(x)^3-4}}\right )}{\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*}