56.17.8 problem Ex 8

Internal problem ID [14189]
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 8
Date solved : Thursday, October 02, 2025 at 09:25:46 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

\begin{align*} \left (x^{2}+1\right ) {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2}-1&=0 \end{align*}

✓ Maple. Time used: 0.081 (sec). Leaf size: 57

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

\begin{align*} y &= \sqrt {x^{2}+1} \\ y &= -\sqrt {x^{2}+1} \\ y &= c_1 x -\sqrt {-c_1^{2}+1} \\ y &= c_1 x +\sqrt {-c_1^{2}+1} \\ \end{align*}

✓ Mathematica. Time used: 0.094 (sec). Leaf size: 73

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

\begin{align*} y(x)&\to c_1 x-\sqrt {1-c_1{}^2}\\ y(x)&\to c_1 x+\sqrt {1-c_1{}^2}\\ y(x)&\to -\sqrt {x^2+1}\\ y(x)&\to \sqrt {x^2+1} \end{align*}

✗ Sympy

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

Timed Out