Singular points are specific values in differential equations where the solutions may behave unusually. They occur when the coefficients of the differential equation's terms go to zero or infinity. There are two main types of singular points: regular and irregular.
- **Regular Singular Points:** If the coefficients of the lower-order derivatives in a differential equation (after being divided by the leading coefficient) remain well-behaved—meaning they don't approach infinity or become undefined—as you approach the singular point, it's classified as regular.
- **Irregular Singular Points:** Conversely, if these coefficients do become infinite or undefined, it's termed an irregular singular point.
Identifying whether a singular point is regular or irregular is crucial, as the equation's behavior around these points determines the methods for finding solutions. In this particular problem, we identified that at the singular point (\(x = 0\)) , the behavior of the function \(\cot x\) becomes infinite, indicating that \(x = 0\) is an irregular singular point.