18.1 problem Ex 1

Internal problem ID [10222]

Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section: Chapter V, Singular solutions. Article 30. Page 63
Problem number: Ex 1.
ODE order: 1.
ODE degree: 2.

CAS Maple gives this as type [[_homogeneous, class G], _rational, _Clairaut]

Solve \begin {gather*} \boxed {y-y^{\prime } x -\frac {1}{y^{\prime }}=0} \end {gather*}

Solution by Maple

Time used: 0.031 (sec). Leaf size: 27

dsolve(y(x)=diff(y(x),x)*x+1/diff(y(x),x),y(x), singsol=all)
 

\begin{align*} y \relax (x ) = -2 \sqrt {x} \\ y \relax (x ) = 2 \sqrt {x} \\ y \relax (x ) = x c_{1}+\frac {1}{c_{1}} \\ \end{align*}

Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 41

DSolve[y[x]==y'[x]*x+1/y'[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_1 x+\frac {1}{c_1} \\ y(x)\to \text {Indeterminate} \\ y(x)\to -2 \sqrt {x} \\ y(x)\to 2 \sqrt {x} \\ \end{align*}