56.17.1 problem Ex 1

Internal problem ID [14182]
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 1
Date solved : Thursday, October 02, 2025 at 09:25:13 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.104 (sec). Leaf size: 54

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

\begin{align*} y &= -a \\ y &= a \\ y &= \sqrt {-c_1^{2}+2 c_1 x +a^{2}-x^{2}} \\ y &= -\sqrt {\left (a +x -c_1 \right ) \left (c_1 +a -x \right )} \\ \end{align*}

✓ Mathematica. Time used: 0.141 (sec). Leaf size: 101

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

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

✓ Sympy. Time used: 2.590 (sec). Leaf size: 85

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

\[ \left [ y{\left (x \right )} = - \sqrt {- C_{1}^{2} + 2 C_{1} x + a^{2} - x^{2}}, \ y{\left (x \right )} = \sqrt {- C_{1}^{2} + 2 C_{1} x + a^{2} - x^{2}}, \ y{\left (x \right )} = - \sqrt {- C_{1}^{2} - 2 C_{1} x + a^{2} - x^{2}}, \ y{\left (x \right )} = \sqrt {- C_{1}^{2} - 2 C_{1} x + a^{2} - x^{2}}\right ] \]