Internal problem ID [6868]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th
edition. 1997.
Section: CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number: 4.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational]
\[ \boxed {4 y^{3} {y^{\prime }}^{2}-4 y^{\prime } x +y=0} \]
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 81
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 )+\int _{}^{\textit {\_Z}}-\frac {2 \left (\textit {\_a}^{4}+\sqrt {-\textit {\_a}^{4}+1}-1\right )}{\textit {\_a} \left (\textit {\_a}^{4}-1\right )}d \textit {\_a} +c_{1} \right ) \sqrt {x} \end{align*}
✓ Solution by Mathematica
Time used: 0.581 (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*}