Internal problem ID [10206]
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]
Solve \begin {gather*} \boxed {x y^{2} \left (y^{\prime }\right )^{2}-y^{3} y^{\prime }+x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (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 \relax (x ) = \sqrt {-2 x} \\ y \relax (x ) = -\sqrt {-2 x} \\ y \relax (x ) = \sqrt {2}\, \sqrt {x} \\ y \relax (x ) = -\sqrt {2}\, \sqrt {x} \\ y \relax (x ) = {\mathrm e}^{\frac {c_{1}}{2}+\frac {\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 \relax (x )}{2}} \\ y \relax (x ) = {\mathrm e}^{-\frac {c_{1}}{2}+\frac {\RootOf \left (x^{2} \left (16 \,{\mathrm e}^{-2 c_{1}} {\mathrm e}^{2 \textit {\_Z}} x^{2}+{\mathrm e}^{2 \textit {\_Z}}-4 \,{\mathrm e}^{-2 c_{1}} {\mathrm e}^{3 \textit {\_Z}} x \right )\right )}{2}+\frac {\ln \relax (x )}{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.032 (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*}