Statement of Frobenius Theorem for Regular (Ordinary) Point

Earlier, we saw a few properties of a power series and some uses also. Presently, we inquire the question, namely, whether an equation of the form

$$y^{\prime\prime} + a(x) y^\prime + b(x) y = f(x), \;\; x \in I$$ (9.3.1)

admits a solution $y$ which has a power series representation around $x \in I.$ In other words, we are interested in looking into an existence of a power series solution of (9.3.1) under certain conditions on $a(x), b(x)$ and $f(x).$ The following is one such result. We omit its proof.

THEOREM 9.3.1   Let $a(x), b(x)$ and $f(x)$ admit a power series representation around a point $x = x_0 \in I,$ with non-zero radius of convergence $r_1, r_2$ and $r_3,$ respectively. Let $R = \min \{r_1, r_2, r_3\}.$ Then the Equation (9.3.1) has a solution $y$ which has a power series representation around $x_0$ with radius of convergence $R.$

Remark 9.3.2   We remind the readers that Theorem 9.3.1 is true for Equations (9.3.1), whenever the coefficient of $y^{\prime\prime}$ is $1.$

Secondly, a point $x_0$ is called an ORDINARY POINT for (9.3.1) if $a(x), b(x)$ and $f(x)$ admit power series expansion (with non-zero radius of convergence) around $x = x_0.$ $\; x_0$ is called a SINGULAR POINT for (9.3.1) if $x_0$ is not an ordinary point for (9.3.1).

The following are some examples for illustration of the utility of Theorem 9.3.1.

EXERCISE 9.3.3  

1. Examine whether the given point $x_0$ is an ordinary point or a singular point for the following differential equations.
   1. $(x-1)y^{\prime\prime} + \sin x y = 0, \; x_0 = 0.$
   2. $y^{\prime\prime} + \frac{\sin x}{x-1} y = 0, \; x_0 = 0.$
   3. Find two linearly independent solutions of
   4. $(1-x^2) y^{\prime\prime} - 2 x y^\prime + n(n+1) y = 0,\; x_0 = 0, \; n$ is a real constant.
2. Show that the following equations admit power series solutions around a given $x_0.$ Also, find the power series solutions if it exists.
   1. $y^{\prime\prime} + y = 0, \; x_0 = 0.$
   2. $x y^{\prime\prime} + y = 0, \; x_0 = 0.$
   3. $y^{\prime\prime} + 9 y = 0, \; x_0 = 0.$