69.4.6 problem 51

Internal problem ID [17995]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 4. Equations with variables separable and equations reducible to them. Exercises page 38
Problem number : 51
Date solved : Thursday, October 02, 2025 at 02:32:40 PM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 5

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

\[ y = 1 \]

✓ Mathematica. Time used: 4.056 (sec). Leaf size: 32

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

\begin{align*} y(x)&\to 1\\ y(x)&\to \sqrt {x^2+2 \sqrt {1-x^2}-1} \end{align*}

✓ Sympy. Time used: 1.016 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = \sqrt {x^{2} + 2 \sqrt {1 - x^{2}} - 1} \]