62.17.3 problem Ex 3

Internal problem ID [12832]
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 28. Summary. Page 59
Problem number : Ex 3
Date solved : Monday, March 31, 2025 at 07:20:28 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+1&=0 \end{align*}

✓ Maple. Time used: 0.338 (sec). Leaf size: 53

ode:=x^3*diff(y(x),x)^2+x^2*y(x)*diff(y(x),x)+1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {2}{\sqrt {x}} \\ y &= \frac {2}{\sqrt {x}} \\ y &= \frac {c_1^{2} x +4}{2 c_1 x} \\ y &= \frac {c_1^{2}+4 x}{2 c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 1.766 (sec). Leaf size: 77

ode=x^3*(D[y[x],x])^2+x^2*y[x]*D[y[x],x]+1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {e^{-\frac {c_1}{2}} \left (x+4 e^{c_1}\right )}{2 x} \\ y(x)\to \frac {e^{-\frac {c_1}{2}} \left (x+4 e^{c_1}\right )}{2 x} \\ y(x)\to -\frac {2}{\sqrt {x}} \\ y(x)\to \frac {2}{\sqrt {x}} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), x)**2 + x**2*y(x)*Derivative(y(x), x) + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**2*y(x) + sqrt(x**3*(x*y(x)**2 - 4)))/(2*x**3) cannot be solved by the factorable group method