Internal problem ID [6038]
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: 13.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational]
Solve \begin {gather*} \boxed {4 y^{3} \left (y^{\prime }\right )^{2}-4 x y^{\prime }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.562 (sec). Leaf size: 83
dsolve(4*y(x)^3*diff(y(x),x)^2-4*x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \sqrt {-x} \\ y \relax (x ) = -\sqrt {-x} \\ y \relax (x ) = \sqrt {x} \\ y \relax (x ) = -\sqrt {x} \\ y \relax (x ) = 0 \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {-2 \textit {\_a}^{4}+2 \sqrt {-\textit {\_a}^{4}+1}+2}{\textit {\_a} \left (\textit {\_a}^{4}-1\right )}d \textit {\_a} +c_{1}\right ) \sqrt {x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.933 (sec). Leaf size: 282
DSolve[4*y[x]^3*(y'[x])^2-4*x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -e^{\frac {c_1}{4}} \sqrt [4]{e^{c_1}-2 i x} \\ y(x)\to -i e^{\frac {c_1}{4}} \sqrt [4]{e^{c_1}-2 i x} \\ y(x)\to i e^{\frac {c_1}{4}} \sqrt [4]{e^{c_1}-2 i x} \\ y(x)\to e^{\frac {c_1}{4}} \sqrt [4]{e^{c_1}-2 i x} \\ y(x)\to -e^{\frac {c_1}{4}} \sqrt [4]{2 i x+e^{c_1}} \\ y(x)\to -i e^{\frac {c_1}{4}} \sqrt [4]{2 i x+e^{c_1}} \\ y(x)\to i e^{\frac {c_1}{4}} \sqrt [4]{2 i x+e^{c_1}} \\ y(x)\to e^{\frac {c_1}{4}} \sqrt [4]{2 i x+e^{c_1}} \\ y(x)\to 0 \\ y(x)\to -\sqrt {x} \\ y(x)\to -i \sqrt {x} \\ y(x)\to i \sqrt {x} \\ y(x)\to \sqrt {x} \\ \end{align*}