19.1 problem Ex 5

Internal problem ID [10224]

Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section: Chapter V, Singular solutions. Article 32. Page 69
Problem number: Ex 5.
ODE order: 1.
ODE degree: 2.

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

Solve \begin {gather*} \boxed {x^{2} \left (y^{\prime }\right )^{2}-2 \left (y x -2\right ) y^{\prime }+y^{2}=0} \end {gather*}

Solution by Maple

Time used: 0.015 (sec). Leaf size: 35

dsolve(x^2*diff(y(x),x)^2-2*(x*y(x)-2)*diff(y(x),x)+y(x)^2=0,y(x), singsol=all)
 

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

Solution by Mathematica

Time used: 0.268 (sec). Leaf size: 43

DSolve[x^2*(y'[x])^2-2*(x*y[x]-2)*y'[x]+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {4 (-x+c_1)}{c_1{}^2} \\ y(x)\to -\frac {4 (x+c_1)}{c_1{}^2} \\ y(x)\to 0 \\ y(x)\to \frac {1}{x} \\ \end{align*}