Internal problem ID [11230]
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 27. Clairaut equation. Page 56
Problem number: Ex 5.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational]
\[ \boxed {x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }=-x} \]
✓ Solution by Maple
Time used: 0.547 (sec). Leaf size: 141
dsolve(x*y(x)^2*diff(y(x),x)^2-y(x)^3*diff(y(x),x)+x=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \sqrt {-2 x} y \left (x \right ) = -\sqrt {-2 x} y \left (x \right ) = \sqrt {x}\, \sqrt {2} y \left (x \right ) = -\sqrt {x}\, \sqrt {2} y \left (x \right ) = {\mathrm e}^{\frac {c_{1}}{2}+\frac {\operatorname {RootOf}\left (16 x \,{\mathrm e}^{2 c_{1}} {\mathrm e}^{2 \textit {\_Z}}+{\mathrm e}^{2 \textit {\_Z}} x^{3}-4 \,{\mathrm e}^{2 c_{1}} {\mathrm e}^{3 \textit {\_Z}}\right )}{2}-\frac {\ln \left (x \right )}{2}} y \left (x \right ) = {\mathrm e}^{-\frac {c_{1}}{2}+\frac {\operatorname {RootOf}\left (x^{2} \left (16 \,{\mathrm e}^{-2 c_{1}} {\mathrm e}^{2 \textit {\_Z}} x^{2}-4 \,{\mathrm e}^{-2 c_{1}} {\mathrm e}^{3 \textit {\_Z}} x +{\mathrm e}^{2 \textit {\_Z}}\right )\right )}{2}+\frac {\ln \left (x \right )}{2}} \end{align*}
✓ Solution by Mathematica
Time used: 6.367 (sec). Leaf size: 187
DSolve[x*y[x]^2*(y'[x])^2-y[x]^3*y'[x]+x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-2 e^{-c_1} x^2-\frac {e^{c_1}}{2}} y(x)\to \sqrt {-2 e^{-c_1} x^2-\frac {e^{c_1}}{2}} y(x)\to -\frac {\sqrt {4 e^{-c_1} x^2+e^{c_1}}}{\sqrt {2}} y(x)\to \frac {\sqrt {4 e^{-c_1} x^2+e^{c_1}}}{\sqrt {2}} y(x)\to -\sqrt {2} \sqrt {x} y(x)\to -i \sqrt {2} \sqrt {x} y(x)\to i \sqrt {2} \sqrt {x} y(x)\to \sqrt {2} \sqrt {x} \end{align*}