83.22.24 problem 24

Internal problem ID [19207]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 24
Date solved : Monday, March 31, 2025 at 06:56:44 PM
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 \end{align*}

✓ Maple. Time used: 0.082 (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; 
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: 0.106 (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; 
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