44.5.27 problem 27

Internal problem ID [7089]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 27
Date solved : Sunday, March 30, 2025 at 11:40:15 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=\frac {\sqrt {3}}{2} \end{align*}

✓ Maple. Time used: 0.223 (sec). Leaf size: 11

ode:=(1-y(x)^2)^(1/2)-(-x^2+1)^(1/2)*diff(y(x),x) = 0; 
ic:=y(0) = 1/2*3^(1/2); 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \sin \left (\arcsin \left (x \right )+\frac {\pi }{3}\right ) \]

✓ Mathematica. Time used: 0.2 (sec). Leaf size: 16

ode=Sqrt[1-y[x]^2]-Sqrt[1-x^2]*D[y[x],x]==0; 
ic={y[0]==Sqrt[3]/2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \cos \left (\frac {\pi }{6}-\arcsin (x)\right ) \]

✗ Sympy

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 = {y(0): sqrt(3)/2} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : Initial conditions produced too many solutions for constants