85.34.2 problem 2

Internal problem ID [22710]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. B Exercises at page 67
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:11:23 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (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: 0.124 (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: 0.171 (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)
 

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