41.1.4 problem 4

Internal problem ID [8671]
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 : 4
Date solved : Tuesday, September 30, 2025 at 05:40:36 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 9

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

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

✓ Mathematica. Time used: 0.111 (sec). Leaf size: 29

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

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

✓ Sympy. Time used: 0.145 (sec). Leaf size: 8

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

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