29.33.9 problem 971

Internal problem ID [5548]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 971
Date solved : Tuesday, March 04, 2025 at 09:59:36 PM
CAS classification : [_quadrature]

\begin{align*} y^{2} {y^{\prime }}^{2}-a^{2}+y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.128 (sec). Leaf size: 54

ode:=y(x)^2*diff(y(x),x)^2-a^2+y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.222 (sec). Leaf size: 101

ode=y[x]^2 (D[y[x],x])^2-a^2 +y[x]^2==0; 
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: 4.015 (sec). Leaf size: 85

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

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