17.10 problem Ex 10

Internal problem ID [10220]

Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article 28. Summary. Page 59
Problem number: Ex 10.
ODE order: 1.
ODE degree: 2.

CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]

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

Solution by Maple

Time used: 0.094 (sec). Leaf size: 91

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

\begin{align*} y \relax (x ) = -\frac {i x^{2}}{2} \\ y \relax (x ) = \frac {i x^{2}}{2} \\ y \relax (x ) = 0 \\ y \relax (x ) = -\frac {\sqrt {-4 x^{2} c_{1}+c_{1}^{2}}}{4} \\ y \relax (x ) = \frac {\sqrt {-4 x^{2} c_{1}+c_{1}^{2}}}{4} \\ y \relax (x ) = -\frac {2 \sqrt {x^{2} c_{1}+4}}{c_{1}} \\ y \relax (x ) = \frac {2 \sqrt {x^{2} c_{1}+4}}{c_{1}} \\ \end{align*}

Solution by Mathematica

Time used: 0.69 (sec). Leaf size: 223

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

\begin{align*} \text {Solve}\left [\frac {\sqrt {x^6+4 x^2 y(x)^2} \log \left (\sqrt {x^4+4 y(x)^2}+x^2\right )}{2 x \sqrt {x^4+4 y(x)^2}}+\frac {1}{2} \left (1-\frac {\sqrt {x^6+4 x^2 y(x)^2}}{x \sqrt {x^4+4 y(x)^2}}\right ) \log (y(x))=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {1}{2} \left (\frac {\sqrt {x^6+4 x^2 y(x)^2}}{x \sqrt {x^4+4 y(x)^2}}+1\right ) \log (y(x))-\frac {\sqrt {x^6+4 x^2 y(x)^2} \log \left (\sqrt {x^4+4 y(x)^2}+x^2\right )}{2 x \sqrt {x^4+4 y(x)^2}}=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}