62.17.1 problem Ex 1

Internal problem ID [12822]
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 : Wednesday, March 05, 2025 at 08:45:03 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.125 (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 {a^{2}-c_{1}^{2}+2 c_{1} x -x^{2}} \\ y &= -\sqrt {\left (a +x -c_{1} \right ) \left (c_{1} +a -x \right )} \\ \end{align*}

✓ Mathematica. Time used: 0.235 (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: 4.105 (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)
 

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