Question

Determine the singular points of the given differential equation. Classify each singular point as regular or irregular.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{x}}\mathit{y}\mathbf{\text{'}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{3}}}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

Therefore, $x=0$is a regular singular point and$x=1$ is an irregular singular point.

Step-by-step solution

To Find the singular points of the given differential equation

We have;

$y\text{'}\text{'}-\frac{1}{x}y\text{'}+\frac{1}{{(x-1)}^3}y=0$

We aim to find the singular points of the given differential equation and classify them. First, let's assume we have a differential equation with the following form;

$$\begin{gathered} a_1\left(x\right)y\text{'}\text{'}+a_2\left(x\right)y\text{'}+a_3\left(x\right)y=0 \\ y\text{'}\text{'}+\frac{a_2\left(x\right)}{a_1\left(x\right)}y\text{'}+\frac{a_3\left(x\right)}{a_2\left(x\right)}y=0 \\ y\text{'}\text{'}+P\left(x\right)y\text{'}+Q\left(x\right)y=0 \dots \left(1\right) \end{gathered}$$

Where;

$P\left(x\right)=\frac{a_2\left(x\right)}{a_1\left(x\right)}andQ\left(x\right)=\frac{a_3\left(x\right)}{a_1\left(x\right)}\dots \left(2\right)$

Find the ordinary points of differential equation

A point $x=X_0$is said to be an ordinary point of the differential equation, if we have an analytic function at a point, this implies that we can expand this function around that point using Taylor Series, which is calculated by the function's derivatives at that point.

If the function has singular points, you will not be able to find the function's derivatives at that point, and it will blow up.

We will be able to deduce that$P\left(x\right)$and$Q\left(x\right)$are not analytic at the points, which will make the denominator equal to zero, after common factors reduction. Now, we put the given differential equation into the standard form.

$P\left(x\right)=\frac{-1}{x}andQ\left(x\right)=\frac{1}{{(x-1)}^3} \dots \left(3\right)$

Now, we equate the denominator by 0 and find the value which satisfies this condition.

$x=0$and${\left(x-1\right)}^3=0\Rightarrow x=1$

Therefore,$P\left(x\right)$ and$Q\left(x\right)$ are not analytic at$x=0$ and$x=1$ which implies that the differential equation has a singular points at$x=0$ and$x=1$ Now, we need to classify these singular points. We define two functions as follows:

$p\left(x\right)=\left(x-x_0\right)P\left(x\right)andq\left(x\right)={\left(x-x_0\right)}^{2}Q\left(x\right) \dots \left(4\right)$

Where$x_0$ is the singular point.

A singular point$x_0$ is said to be a regular singular point of the differential equation, if underline$p\left(x\right)$ and$q\left(x\right)$ are analytic at$x_0$ otherwise it is irregular singular point.

Final proof

Step 3: Final proof

We are left with classifying each point, at which the differential equation has a singular point. There is a way, by which we can classify the singular points, by taking the limit for (4) as$x$goes to $x_0$. If the limit exists, then the function is analytic at that point. And if the limit doesn't exist, this implies that the function is not analytic at that point.

At $x=0$

We find$p\left(x\right)$and$q\left(x\right)$as follows. From (3) and (4), we get

$$\begin{gathered} \underset{x\to 0}{\lim }p\left(x\right)=\underset{x\to 0}{\lim }(x-0)·\frac{-1}{x} \\ =-1\left[\text{p}\right(\text{x})\text{is analytic at x}=0] \\ \underset{x\to 0}{\lim }q\left(x\right)=\underset{x\to 0}{\lim }{(x-0)}^2·\frac{1}{{(x-1)}^3}\left[\text{q}\right(\text{x})\text{is analytic at x}=0] \\ =0 \end{gathered}$$

Therefore,$x=0$is a regular point.

At $x=1$From (3) and (4), we get:

$$\begin{gathered} \underset{x\to 1}{\lim }p\left(X\right)=\underset{x\to 1}{\lim }(x-1)·\frac{-1}{x} \\ =0[\text{p}(\text{x})\text{is analytic at x}=1] \\ \underset{x\to 1}{\lim }q\left(X\right)=\underset{x\to 1}{\lim }{(x-1)}^2·\frac{1}{{(x-1)}^3} \\ \underset{x\to 1}{=\lim }\frac{1}{(x-1)} \\ =doesn\text{'}t exist\left[\text{q}\right(\text{x})\text{is not analytic at x}=1] \end{gathered}$$

Therefore,$x=1$ is an irregular singular point.

Thus,$x=0$ is a regular singular point and$x=1$ is an irregular singular point.