62.12.2 problem Ex 2

Internal problem ID [12769]
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 2
Date solved : Wednesday, March 05, 2025 at 08:27:34 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 11

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

\[ y = -\sin \left (\arcsin \left (x \right )+c_{1} \right ) \]

✓ Mathematica. Time used: 0.202 (sec). Leaf size: 33

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

\begin{align*} y(x)\to -\sin (\arcsin (x)-c_1) \\ y(x)\to -1 \\ y(x)\to 1 \\ y(x)\to \text {Interval}[\{-1,1\}] \\ \end{align*}

✓ Sympy. Time used: 0.290 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = \sin {\left (C_{1} - \operatorname {asin}{\left (x \right )} \right )} \]