83.46.2 problem Ex 2 page 69

Internal problem ID [19488]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter V. Singular solutions
Problem number : Ex 2 page 69
Date solved : Thursday, March 13, 2025 at 02:40:03 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 54

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

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

✓ Mathematica. Time used: 0.211 (sec). Leaf size: 101

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

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

✓ Sympy. Time used: 3.950 (sec). Leaf size: 85

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

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