Question

Find all singular points of the given equation and determine whether each one is regular or irregular. \(x y^{\prime \prime}+y^{\prime}+(\cot x) y=0\)

Short answer

The singular point for the given differential equation \(x y^{\prime \prime} + y^{\prime} + (\cot x) y = 0\) is \(x=0\). This point is determined to be irregular because as \(x\) approaches \(0\), the coefficient of \(y\), which is \(\cot x\), becomes undefined. Therefore, the coefficient of \(y\) isn't analytic at \(x = 0\), making the singular point irregular.

Step-by-step solution

Identify the Singular Points

We need to identify when the coefficient of the highest derivative becomes zero. In the given equation, the highest derivative is \(y^{\prime \prime}\) and its coefficient is \(x\). So, we should find the values of \(x\) for which the coefficient becomes zero. Set the coefficient equal to zero: \(x=0\) So, there is only one singular point, \(x=0\).

Determine the Regular or Irregular Singular Point

Now that we found the singular point \(x=0\), we need to determine if it is regular or irregular. To do this, we will analyze the behavior of the other coefficients in the differential equation (\(y'\) coefficient and \(y\) coefficient) as \(x\) approaches the singular point \(x=0\). The given differential equation is: \(x y^{\prime \prime} + y^{\prime} + (\cot x)y = 0\) As \(x\) approaches \(0\), the coefficient of \(y\) approaches infinity because the cotangent function, \(\cot x\), becomes undefined at \(x=0\). This means that the coefficient of \(y\) isn't analytic at \(x = 0\). Therefore, the singular point is irregular.

Result

There is one singular point, \(x=0\), and it is an irregular singular point for the given differential equation.