75.4.6 problem 51

Internal problem ID [16708]
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 : Tuesday, January 28, 2025 at 09:18:59 AM
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*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 5

dsolve([x*sqrt(1-y(x)^2)+y(x)*sqrt(1-x^2)*diff(y(x),x)=0,y(0) = 1],y(x), singsol=all)
 

\[ y = 1 \]

✓ Solution by Mathematica

Time used: 3.570 (sec). Leaf size: 32

DSolve[{x*Sqrt[1-y[x]^2]+y[x]*Sqrt[1-x^2]*D[y[x],x]==0,{y[0]==1}},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*}