56.16.5 problem Ex 5

Internal problem ID [14177]
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 : Thursday, October 02, 2025 at 09:22:40 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: 0.137 (sec). Leaf size: 159

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} \\ \left (\ln \left (\frac {y^{2} \left (y^{2}+\operatorname {csgn}\left (y^{2}\right ) \sqrt {y^{4}-4 x^{2}}\right )}{x}\right )+\ln \left (2\right )-2 \ln \left (y\right )\right ) \operatorname {csgn}\left (y^{2}\right )+c_1 -\ln \left (x \right ) &= 0 \\ \left (-\ln \left (\frac {y^{2} \left (y^{2}+\operatorname {csgn}\left (y^{2}\right ) \sqrt {y^{4}-4 x^{2}}\right )}{x}\right )-\ln \left (2\right )+2 \ln \left (y\right )\right ) \operatorname {csgn}\left (y^{2}\right )+c_1 -\ln \left (x \right ) &= 0 \\ \end{align*}

✓ Mathematica. Time used: 1.031 (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