91Ó°ÊÓ

A point ^$x_0$ is called an ordinary point of equation y"+p(x)y'+q(x)y = 0 if both pand qare analytic at ^$x_0$ . If ^$x_0$is not an ordinary point, it is called a singular point of the equation.

The given differential equation is

$({\mathrm{θ}}^2-2)y"+2y\text{'}+\sin \mathrm{θ}y=0$

Dividing the above equation by (${\mathrm{Ï‘}}^2-2$) we get,

$y"+\frac{2}{({\mathrm{θ}}^2-2)}y\text{'}+\frac{\sin \mathrm{θ}}{({\mathrm{θ}}^2-2)}y=0$

On comparing the above equation with y"+p(x)y'+q(x)y=0 , we find that,

P(x)=$\frac{2}{({\mathrm{θ}}^2-2)}$

Q(x)=$\frac{\sin \mathrm{θ}}{({\mathrm{θ}}^2-2)}$

Hence, ^P(x) and ^Q(x) are analytic except, perhaps, when their denominators are zero.

For ^P(x) this occurs at x = $Â\pm \sqrt{2}$.

Since, ^Q(x) has the power series expansion but it analytic except at x=$Â\pm \sqrt{2}$.

Therefore, ^P(x) is analytic except at x =$Â\pm \sqrt{2}$as well as ^Q(x) is analytic except at x =$Â\pm \sqrt{2}$.

The singular point exists in this differential equation for both ^P(x) and ^Q(x) is at x=$Â\pm \sqrt{2}$.