53.3.6 problem 8

Internal problem ID [8468]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 99. Clairaut equation. EXERCISES Page 320
Problem number : 8
Date solved : Monday, January 27, 2025 at 04:06:25 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} x^{8} {y^{\prime }}^{2}+3 x y^{\prime }+9 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 45

dsolve(x^8*diff(y(x),x)^2+3*x*diff(y(x),x)+9*y(x)=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {1}{4 x^{6}} \\ y &= \frac {-x^{3}+c_{1}}{x^{3} c_{1}^{2}} \\ y &= \frac {-x^{3}-c_{1}}{c_{1}^{2} x^{3}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.624 (sec). Leaf size: 130

DSolve[x^8*(D[y[x],x])^2+3*x*D[y[x],x]+9*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [\frac {x \sqrt {4 x^6 y(x)-1} \arctan \left (\sqrt {4 x^6 y(x)-1}\right )}{3 \sqrt {x^2-4 x^8 y(x)}}-\frac {1}{6} \log (y(x))&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {x^2-4 x^8 y(x)} \arctan \left (\sqrt {4 x^6 y(x)-1}\right )}{3 x \sqrt {4 x^6 y(x)-1}}-\frac {1}{6} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}