Internal problem ID [12167]
Book: DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR
PUBLISHERS, Moscow 1969.
Section: Chapter 8. Differential equations. Exercises page 595
Problem number: 98.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Clairaut]
\[ \boxed {y-x y^{\prime }+\frac {1}{{y^{\prime }}^{2}}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 77
dsolve(y(x)=x*diff(y(x),x)-1/diff(y(x),x)^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {3 \left (-2 x^{2}\right )^{\frac {1}{3}}}{2} y \left (x \right ) = -\frac {3 \left (-2 x^{2}\right )^{\frac {1}{3}}}{4}-\frac {3 i \sqrt {3}\, \left (-2 x^{2}\right )^{\frac {1}{3}}}{4} y \left (x \right ) = -\frac {3 \left (-2 x^{2}\right )^{\frac {1}{3}}}{4}+\frac {3 i \sqrt {3}\, \left (-2 x^{2}\right )^{\frac {1}{3}}}{4} y \left (x \right ) = c_{1} x -\frac {1}{c_{1}^{2}} \end{align*}
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 71
DSolve[y[x]==x*y'[x]-1/(y'[x])^2,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*}