41.1.18 problem 18

Internal problem ID [8685]
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 : 18
Date solved : Tuesday, September 30, 2025 at 05:41:08 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 11

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

\[ y = -\sin \left (\arcsin \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.126 (sec). Leaf size: 23

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

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

✓ Sympy. Time used: 0.171 (sec). Leaf size: 8

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

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