Internal problem ID [4242]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {x \sqrt {1-y^{2}}+y \sqrt {1-x^{2}}\, y^{\prime }=0} \end {gather*} With initial conditions \begin {align*} \left [y \left (\frac {1}{2}\right ) = {\frac {1}{2}}\right ] \end {align*}
✓ Solution by Maple
Time used: 0.532 (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)
\[ y \relax (x ) = \sqrt {2 \sqrt {3}\, \sqrt {-x^{2}+1}+x^{2}-3} \]
✓ Solution by Mathematica
Time used: 3.781 (sec). Leaf size: 38
DSolve[{x*Sqrt[1-y[x]^2]+y[x]*Sqrt[1-x^2]*y'[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*}