I am studying the Frobenius method to solve series solution of differential equations around a regular singular point. I am using a book by an Indian author which defines the definitions as follows:
Consider the general homogeneous second order linear differential equation $$y''+P(x)y'+Q(x)y=0$$ Then the point $x_0\in D$ is said to be an ordinary point of the above the given differential equation if $P(x)$ and $Q(x)$ are analytic at $x_0$. If either $P(x)$ or $Q(x)$ fails to be analytic at $x_0$, the point $x_0$ is called a singular point of the given differential equation.
Further a singular point $x_0$ of the given differential equation is said to be a regular singular point if the functions $(x−x_0)P(x)$ and $(x−x_0)^2Q(x)$ are analytic at $x_0$ and irregular otherwise.
Now the following question was asked in my exam:
Question: Determine how many linearly independent solutions does the differential equation $$x^2y''+3\sin(x)y'+y=0$$ have around X=0?
The above questions creates a confusion in my mind:
- Here $P(x)=\frac{3\sin x}{x^2}$ and $Q(x)=\frac{1}{x^2}$ and hence we have function $(x−x_0)P(x)=\frac{3\sin x}{x}$ and $(x−x_0)^2Q(x)=1$ then clearly $\frac{3\sin x}{x}$ is not analytic at $x=0$? So it means by definition 0 is irregular singular point, right? then how can we talk about series solution by Frobenius method around x=0, as we talk Frobenius method about regular singular point.
- Secondly, what is the answer to this question (i.e how many linearly independent solutions does given differential equation has?
Any help would be appreciable!