69.1.19 problem 22

Internal problem ID [14093]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 22
Date solved : Wednesday, March 05, 2025 at 10:31:26 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 9

ode:=(-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: 0.19 (sec). Leaf size: 29

ode=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 -1 \\ y(x)\to 1 \\ y(x)\to \text {Interval}[\{-1,1\}] \\ \end{align*}

✓ Sympy. Time used: 0.294 (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)
 

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