Internal problem ID [6122]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number: 14.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class G`]]
\[ \boxed {16 x {y^{\prime }}^{2}+8 y y^{\prime }+y^{6}=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 103
dsolve(16*x*diff(y(x),x)^2+8*y(x)*diff(y(x),x)+y(x)^6=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {1}{x^{\frac {1}{4}}} \\ y \left (x \right ) = -\frac {1}{x^{\frac {1}{4}}} \\ y \left (x \right ) = -\frac {i}{x^{\frac {1}{4}}} \\ y \left (x \right ) = \frac {i}{x^{\frac {1}{4}}} \\ y \left (x \right ) = 0 \\ y \left (x \right ) = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_{1} +4 \left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} \sqrt {-\textit {\_a}^{4}+1}}d \textit {\_a} \right )\right )}{x^{\frac {1}{4}}} \\ y \left (x \right ) = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_{1} -4 \left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} \sqrt {-\textit {\_a}^{4}+1}}d \textit {\_a} \right )\right )}{x^{\frac {1}{4}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.664 (sec). Leaf size: 171
DSolve[16*x*(y'[x])^2+8*y[x]*y'[x]+y[x]^6==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {2} e^{\frac {c_1}{4}}}{\sqrt {x+e^{c_1}}} \\ y(x)\to -\frac {i \sqrt {2} e^{\frac {c_1}{4}}}{\sqrt {x+e^{c_1}}} \\ y(x)\to \frac {i \sqrt {2} e^{\frac {c_1}{4}}}{\sqrt {x+e^{c_1}}} \\ y(x)\to \frac {\sqrt {2} e^{\frac {c_1}{4}}}{\sqrt {x+e^{c_1}}} \\ y(x)\to 0 \\ y(x)\to -\frac {1}{\sqrt [4]{x}} \\ y(x)\to -\frac {i}{\sqrt [4]{x}} \\ y(x)\to \frac {i}{\sqrt [4]{x}} \\ y(x)\to \frac {1}{\sqrt [4]{x}} \\ \end{align*}