Internal problem ID [6128]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number: 20.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {x^{6} \left (y^{\prime }\right )^{2}-16 y-8 x y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.281 (sec). Leaf size: 141
dsolve(x^6*diff(y(x),x)^2=8*(2*y(x)+x*diff(y(x),x)),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {1}{x^{4}} \\ y \relax (x ) = \frac {-x^{4}+2 c_{1} \left (i x^{2}+c_{1}\right )-2 c_{1}^{2}}{c_{1}^{2} x^{4}} \\ y \relax (x ) = \frac {-x^{4}+2 c_{1} \left (-i x^{2}+c_{1}\right )-2 c_{1}^{2}}{c_{1}^{2} x^{4}} \\ y \relax (x ) = \frac {-x^{4}-2 c_{1} \left (i x^{2}-c_{1}\right )-2 c_{1}^{2}}{c_{1}^{2} x^{4}} \\ y \relax (x ) = \frac {-x^{4}-2 c_{1} \left (-i x^{2}-c_{1}\right )-2 c_{1}^{2}}{c_{1}^{2} x^{4}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.97 (sec). Leaf size: 122
DSolve[x^6*y'[x]^2==8*(2*y[x]+x*y'[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {x \sqrt {x^4 y(x)+1} \tanh ^{-1}\left (\sqrt {x^4 y(x)+1}\right )}{2 \sqrt {x^6 y(x)+x^2}}-\frac {1}{4} \log (y(x))=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {x \sqrt {x^4 y(x)+1} \tanh ^{-1}\left (\sqrt {x^4 y(x)+1}\right )}{2 \sqrt {x^6 y(x)+x^2}}-\frac {1}{4} \log (y(x))=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}