Internal problem ID [6033]
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.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational]
\[ \boxed {3 x^{4} {y^{\prime }}^{2}-y^{\prime } x -y=0} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 147
dsolve(3*x^4*diff(y(x),x)^2-x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = -\frac {1}{12 x^{2}} \\ y \left (x \right ) = \frac {-c_{1} \left (-c_{1} +2 i x \sqrt {3}\right )-c_{1}^{2}-6 x^{2}}{6 x^{2} c_{1}^{2}} \\ y \left (x \right ) = \frac {-c_{1} \left (-c_{1} -2 i x \sqrt {3}\right )-c_{1}^{2}-6 x^{2}}{6 x^{2} c_{1}^{2}} \\ y \left (x \right ) = \frac {c_{1} \left (c_{1} +2 i x \sqrt {3}\right )-6 x^{2}-c_{1}^{2}}{6 c_{1}^{2} x^{2}} \\ y \left (x \right ) = \frac {c_{1} \left (c_{1} -2 i x \sqrt {3}\right )-6 x^{2}-c_{1}^{2}}{6 c_{1}^{2} x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.494 (sec). Leaf size: 123
DSolve[3*x^4*(y'[x])^2-x*y'[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*}