Internal problem ID [6791]
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]
\[ \boxed {4 y^{3} {y^{\prime }}^{2}-4 x y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.281 (sec). Leaf size: 85
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 \left (x \right ) &= \sqrt {-x} \\ y \left (x \right ) &= -\sqrt {-x} \\ y \left (x \right ) &= \sqrt {x} \\ y \left (x \right ) &= -\sqrt {x} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+2 \left (\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{4}-\sqrt {-\textit {\_a}^{4}+1}-1}{\textit {\_a} \left (\textit {\_a}^{4}-1\right )}d \textit {\_a} \right )+c_{1} \right ) \sqrt {x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.587 (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*}