89.6.31 problem 33 first IC

Internal problem ID [24414]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Miscellaneous Exercises at page 45
Problem number : 33 first IC
Date solved : Thursday, October 02, 2025 at 10:26:13 PM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.100 (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,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.106 (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]
 

\begin{align*} y(x)&\to \cos \left (\frac {\pi }{6}-\arcsin (x)\right ) \end{align*}

✗ 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