56.12.1 problem Ex 1

Internal problem ID [14128]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 19. Summary. Page 29
Problem number : Ex 1
Date solved : Thursday, October 02, 2025 at 09:15:22 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 40

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

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

✓ Mathematica. Time used: 3.607 (sec). Leaf size: 77

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

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

✓ Sympy. Time used: 0.922 (sec). Leaf size: 49

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

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