54.2.12 problem 14

Internal problem ID [8544]
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
Date solved : Monday, January 27, 2025 at 04:13:28 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} 16 x {y^{\prime }}^{2}+8 y^{\prime } y+y^{6}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 101

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 &= \frac {1}{x^{{1}/{4}}} \\ y &= -\frac {1}{x^{{1}/{4}}} \\ y &= -\frac {i}{x^{{1}/{4}}} \\ y &= \frac {i}{x^{{1}/{4}}} \\ y &= 0 \\ y &= \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^{{1}/{4}}} \\ y &= \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^{{1}/{4}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.608 (sec). Leaf size: 171

DSolve[16*x*(D[y[x],x])^2+8*y[x]*D[y[x],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*}