62.16.1 problem Ex 1

Internal problem ID [12821]
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 27. Clairaut equation. Page 56
Problem number : Ex 1
Date solved : Monday, March 31, 2025 at 07:11:43 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

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

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

ode=(D[y[x],x]*x-y[x])^2==(D[y[x],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 {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