65.8.39 problem 13 (c)

Internal problem ID [15760]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.4.4, page 115
Problem number : 13 (c)
Date solved : Thursday, October 02, 2025 at 10:27:24 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&={\frac {1}{2}} \\ \end{align*}

✓ Maple. Time used: 0.090 (sec). Leaf size: 14

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

\[ y = \sin \left (\frac {x^{2}}{2}+\frac {\pi }{6}\right ) \]

✓ Mathematica. Time used: 0.095 (sec). Leaf size: 17

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

\begin{align*} y(x)&\to \sin \left (\frac {1}{6} \left (3 x^2+\pi \right )\right ) \end{align*}

✗ Sympy

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

NotImplementedError : Initial conditions produced too many solutions for constants