Internal problem ID [6056]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 99. Clairaut’s equation. EXERCISES Page
320
Problem number: 17.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Clairaut]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{3}-y \left (y^{\prime }\right )^{2}+1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.375 (sec). Leaf size: 80
dsolve(x*diff(y(x),x)^3-y(x)*diff(y(x),x)^2+1=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{2} \\ y \relax (x ) = -\frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4}-\frac {3 i \sqrt {3}\, 2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4} \\ y \relax (x ) = -\frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4}+\frac {3 i \sqrt {3}\, 2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4} \\ y \relax (x ) = c_{1} x +\frac {1}{c_{1}^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 69
DSolve[x*(y'[x])^3-y[x]*(y'[x])^2+1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x+\frac {1}{c_1{}^2} \\ y(x)\to 3 \left (-\frac {1}{2}\right )^{2/3} x^{2/3} \\ y(x)\to \frac {3 x^{2/3}}{2^{2/3}} \\ y(x)\to -\frac {3 \sqrt [3]{-1} x^{2/3}}{2^{2/3}} \\ \end{align*}