53.2.2 problem 9

Internal problem ID [8455]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 97. The p-discriminant equation. EXERCISES Page 314
Problem number : 9
Date solved : Monday, January 27, 2025 at 04:02:08 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} 3 x^{4} {y^{\prime }}^{2}-x y^{\prime }-y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.042 (sec). Leaf size: 93

dsolve(3*x^4*diff(y(x),x)^2-x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
 

\begin{align*} y &= -\frac {1}{12 x^{2}} \\ y &= \frac {-i \sqrt {3}\, c_{1} -3 x}{3 x \,c_{1}^{2}} \\ y &= \frac {i \sqrt {3}\, c_{1} -3 x}{3 x \,c_{1}^{2}} \\ y &= \frac {i \sqrt {3}\, c_{1} -3 x}{3 x \,c_{1}^{2}} \\ y &= \frac {-i \sqrt {3}\, c_{1} -3 x}{3 x \,c_{1}^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.535 (sec). Leaf size: 123

DSolve[3*x^4*(D[y[x],x])^2-x*D[y[x],x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [-\frac {x \sqrt {12 x^2 y(x)+1} \text {arctanh}\left (\sqrt {12 x^2 y(x)+1}\right )}{\sqrt {12 x^4 y(x)+x^2}}-\frac {1}{2} \log (y(x))&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {x \sqrt {12 x^2 y(x)+1} \text {arctanh}\left (\sqrt {12 x^2 y(x)+1}\right )}{\sqrt {12 x^4 y(x)+x^2}}-\frac {1}{2} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}