77.1.63 problem 82 (page 120)

Internal problem ID [17882]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 82 (page 120)
Date solved : Monday, March 31, 2025 at 04:47:59 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

✓ Maple. Time used: 0.080 (sec). Leaf size: 57

ode:=diff(y(x),x)^2*(x^2-1)-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: 0.11 (sec). Leaf size: 73

ode=D[y[x],x]^2*(x^2-1)-2*D[y[x],x]*x*y[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 {1-x^2} \\ y(x)\to \sqrt {1-x^2} \\ \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