81.8.1 problem 9-1

Internal problem ID [21584]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 9. Clairaut Equation. Page 133.
Problem number : 9-1
Date solved : Thursday, October 02, 2025 at 07:58:40 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

✓ Maple. Time used: 0.054 (sec). Leaf size: 57

ode:=(x*diff(y(x),x)-y(x))^2-diff(y(x),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: 0.077 (sec). Leaf size: 73

ode=(x*D[y[x],x]-y[x] )^2-D[y[x],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((x*Derivative(y(x), x) - y(x))**2 - Derivative(y(x), x)**2 - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out