54.2.17 problem 20

Internal problem ID [8549]
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
Date solved : Monday, January 27, 2025 at 04:13:57 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 85

dsolve(x^6*diff(y(x),x)^2=8*(2*y(x)+x*diff(y(x),x)),y(x), singsol=all)
 

\begin{align*} y &= -\frac {1}{x^{4}} \\ y &= \frac {2 c_{1} i-x^{2}}{x^{2} c_{1}^{2}} \\ y &= \frac {-2 c_{1} i-x^{2}}{x^{2} c_{1}^{2}} \\ y &= \frac {-2 c_{1} i-x^{2}}{x^{2} c_{1}^{2}} \\ y &= \frac {2 c_{1} i-x^{2}}{x^{2} c_{1}^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.561 (sec). Leaf size: 122

DSolve[x^6*D[y[x],x]^2==8*(2*y[x]+x*D[y[x],x]),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [-\frac {x \sqrt {x^4 y(x)+1} \text {arctanh}\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} \text {arctanh}\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*}