62.16.5 problem Ex 5

Internal problem ID [12825]
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 : Monday, March 31, 2025 at 07:17:01 AM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.379 (sec). Leaf size: 131

ode:=x*y(x)^2*diff(y(x),x)^2-y(x)^3*diff(y(x),x)+x = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {2}\, \sqrt {-x} \\ y &= -\sqrt {2}\, \sqrt {-x} \\ y &= \sqrt {2}\, \sqrt {x} \\ y &= -\sqrt {2}\, \sqrt {x} \\ y &= \frac {{\mathrm e}^{\frac {c_1}{2}+\frac {\operatorname {RootOf}\left (\left (-x^{3}-16 x \,{\mathrm e}^{2 c_1}+4 \,{\mathrm e}^{\textit {\_Z} +2 c_1}\right ) {\mathrm e}^{2 \textit {\_Z}}\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 c_1}+4 x \,{\mathrm e}^{\textit {\_Z} -2 c_1}-1\right ) {\mathrm e}^{2 \textit {\_Z}}\right )}{2}} \\ \end{align*}

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

ode=x*y[x]^2*(D[y[x],x])^2-y[x]^3*D[y[x],x]+x==0; 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x)**2*Derivative(y(x), x)**2 + x - y(x)**3*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out