62.16.5 problem Ex 5

Internal problem ID [12896]
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
Date solved : Tuesday, January 28, 2025 at 04:37:22 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }+x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.279 (sec). Leaf size: 140

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 &= \sqrt {2}\, \sqrt {-x} \\ y &= -\sqrt {2}\, \sqrt {-x} \\ y &= \sqrt {x}\, \sqrt {2} \\ y &= -\sqrt {x}\, \sqrt {2} \\ y &= \frac {{\mathrm e}^{\frac {c_{1}}{2}+\frac {\operatorname {RootOf}\left (16 x \,{\mathrm e}^{2 \textit {\_Z} +2 c_{1}}+x^{3} {\mathrm e}^{2 \textit {\_Z}}-4 \,{\mathrm e}^{3 \textit {\_Z} +2 c_{1}}\right )}{2}}}{\sqrt {x}} \\ y &= \sqrt {x}\, {\mathrm e}^{-\frac {c_{1}}{2}+\frac {\operatorname {RootOf}\left (x^{2} \left (16 x^{2} {\mathrm e}^{2 \textit {\_Z} -2 c_{1}}-4 \,{\mathrm e}^{3 \textit {\_Z} -2 c_{1}} x +{\mathrm e}^{2 \textit {\_Z}}\right )\right )}{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.061 (sec). Leaf size: 187

DSolve[x*y[x]^2*(D[y[x],x])^2-y[x]^3*D[y[x],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*}