78.1.3 problem 1.c

Internal problem ID [20929]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 1, First order ODEs. Problems section 1.5
Problem number : 1.c
Date solved : Thursday, October 02, 2025 at 06:49:20 PM
CAS classification : [_separable]

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

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

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

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

✓ Mathematica. Time used: 0.119 (sec). Leaf size: 29

ode=D[y[x],x]==Sqrt[1-y[x]^2]/x; 
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: 0.137 (sec). Leaf size: 8

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

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