82.18.12 problem Ex. 13

Internal problem ID [18755]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Examples on chapter III. page 38
Problem number : Ex. 13
Date solved : Thursday, March 13, 2025 at 12:49:05 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 51

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

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

✓ Mathematica. Time used: 0.141 (sec). Leaf size: 93

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

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

✓ Sympy. Time used: 2.717 (sec). Leaf size: 78

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

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