35.2.2 problem 2

Internal problem ID [6094]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 2. Separable equations. page 398
Problem number : 2
Date solved : Monday, January 27, 2025 at 01:35:43 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 (\frac {1}{2}\right )&={\frac {1}{2}} \end{align*}

✓ Solution by Maple

Time used: 0.500 (sec). Leaf size: 25

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

\[ y = \sqrt {2 \sqrt {3}\, \sqrt {-x^{2}+1}+x^{2}-3} \]

✓ Solution by Mathematica

Time used: 3.615 (sec). Leaf size: 38

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

\begin{align*} y(x)\to \sqrt {x^2} \\ y(x)\to \sqrt {x^2+2 \sqrt {3-3 x^2}-3} \\ \end{align*}