Question: In Problems 1–10, determine all the singular points of the given differential equation.

3.$\left({\theta }^2-2\right)y"+2y\text{'}+\left(\sin \theta \right)y=0$

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

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

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

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

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

P(x)=$\frac{2}{({\theta }^2-2)}$

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

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

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

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

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

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