9.8.37 problem 39

Internal problem ID [3040]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 39
Date solved : Tuesday, September 30, 2025 at 06:24:18 AM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.183 (sec). Leaf size: 22

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

\[ y = \frac {x^{2}}{4}-\frac {\sqrt {-x^{2}+1}}{2}+\frac {1}{2} \]

✓ Mathematica. Time used: 0.14 (sec). Leaf size: 53

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

\begin{align*} y(x)&\to \frac {1}{4} \left (x^2-2 \sqrt {1-x^2}+2\right )\\ y(x)&\to \frac {1}{4} \left (x^2+6 \sqrt {1-x^2}-6\right ) \end{align*}

✗ Sympy

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

NotImplementedError : Initial conditions produced too many solutions for constants