47.1.19 problem 19

Internal problem ID [7400]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number : 19
Date solved : Sunday, March 30, 2025 at 11:57:26 AM
CAS classification : [_separable]

\begin{align*} 2 x \sqrt {1-y^{2}}+y y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

ode:=2*x*(1-y(x)^2)^(1/2)+y(x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ c_1 +x^{2}+\frac {\left (y-1\right ) \left (y+1\right )}{\sqrt {1-y^{2}}} = 0 \]

✓ Mathematica. Time used: 0.305 (sec). Leaf size: 69

ode=2*x*Sqrt[1-y[x]^2]+y[x]*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\sqrt {-x^4+2 c_1 x^2+1-c_1{}^2} \\ y(x)\to \sqrt {-x^4+2 c_1 x^2+1-c_1{}^2} \\ y(x)\to -1 \\ y(x)\to 1 \\ \end{align*}

✓ Sympy. Time used: 1.232 (sec). Leaf size: 42

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

\[ \left [ y{\left (x \right )} = - \sqrt {- C_{1}^{2} + 2 C_{1} x^{2} - x^{4} + 1}, \ y{\left (x \right )} = \sqrt {- C_{1}^{2} + 2 C_{1} x^{2} - x^{4} + 1}\right ] \]