54.2.5 problem 5

Internal problem ID [8537]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number : 5
Date solved : Monday, January 27, 2025 at 04:11:19 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} x^{6} {y^{\prime }}^{2}-2 x y^{\prime }-4 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.697 (sec). Leaf size: 85

dsolve(x^6*diff(y(x),x)^2-2*x*diff(y(x),x)-4*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.556 (sec). Leaf size: 128

DSolve[x^6*(D[y[x],x])^2-2*x*D[y[x],x]-4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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